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  • richardmitnick 8:44 am on April 27, 2023 Permalink | Reply
    Tags: "To improve robots look to birds", , Developing reconfigurable robotic fixtures with multiple robots that can work well together, , , Mathematics, ME researchers improved a mathematical model describing how birds flock together – which could be applied to build robots that move in the same way., , , , The flocking behavior of birds is sometimes referred to as a "murmuration"., The new model shows that delayed self-reinforcement can reduce distortion during information propagation even in noisy environments., The new model was also inspired by the dispersive properties of water waves which are similar to how birds flock., The reconfigurable robotic system can handle a variety of parts and tasks., , To avoid predators flocks of starling birds can perform parallel sharp turns.   

    From The Department of Mechanical Engineering In The College of Engineering At The University of Washington : “To improve robots look to birds” 

    From The Department of Mechanical Engineering


    The College of Engineering


    The University of Washington

    4.10.23 [Just today in social media.]
    Lyra Fontaine

    ME researchers improved a mathematical model describing how birds flock together – which could be applied to build robots that move in the same way.

    ME researchers recently developed an improved mathematical model to describe how birds flock together while suppressing unwanted noise. Photo credit: Rhys Kentish.

    When the light turns green in an intersection, there’s a delay between when the driver in the first car sees the light change and moves forward, and when the driver in the eighth car in line moves forward. Even when the vehicles are autonomous or self-driving, the cars don’t start moving at the same time. More vehicles would be able to cross the intersection at the same light if they all “knew” that the light was turning green, started together and continued moving in the same way.

    Traffic is one area studied by ME researchers, who recently came up with an improved mathematical model to describe how birds flock together while suppressing unwanted noise. This algorithm could be applied to build robots that work together better – whether they’re self-driving cars that can start driving at the same time, or small robots working in a large group to carry an item without stretching or breaking it.

    “The applications are very diverse,” says ME Professor Santosh Devasia.

    Devasia co-authored a paper recently published in PNAS [below]. ME Professor Emeritus James Riley and ME alum Anuj Tiwari (Ph.D., ‘22) are also co-authors.

    Inspired by nature

    In swarms found in nature, information propagates without distortion. For example, to avoid predators, flocks of starling birds can perform parallel sharp turns. This flocking behavior is sometimes referred to as a “murmuration”. Devasia’s model takes into account that each bird adjusts its actions based on observations from its neighbors as well as its own previous actions, a concept he calls delayed self-reinforcement. He has since improved upon his own model.

    Now, the model shows that delayed self-reinforcement can reduce distortion during information propagation, even in noisy environments. This new model could be used to improve cohesion in engineered networks, such as autonomous drone formations and in traffic.

    The algorithm reflects how noise – such as the flapping noises of birds’ wings, or noise from cars moving – can be suppressed so that birds know where to fly next or the autonomous vehicle can “understand” its next action.

    “Our new method removes the high-frequency noise,” Devasia says. “You don’t want to follow the noise; you want to follow the motion.”

    The model is a more accurate representation of how information is propagated in birds. Devasia became interested in the subject from learning about how flocks of starlings fly in gigantic swarms and maneuver together without centralized communication from a leader.

    “Among birds, information doesn’t decay as it propagates,” Devasia says. “We’re trying to figure out why and change our mathematical models to reflect this behavior, because if you can do that, you can get autonomous vehicles and multiple robots to work together.”

    The new model was also inspired by the dispersive properties of water waves, which are similar to how birds flock.

    “It is often quite useful in engineering and mathematics to take advantage of how the results in one field of interest might directly apply to another, such as the cross-fertilization of one field with information from another,” says Riley, a fluid dynamicist who has studied wave dispersion extensively in problems of hydrodynamic stability, water waves and internal gravity waves.

    Applying the algorithm

    Devasia’s lab is applying the algorithm to develop reconfigurable robotic fixtures with multiple robots that can work well together instead of traditional fixtures to hold and manipulate parts where each task requires a new specialized fixture. Instead, the same reconfigurable robotic system can handle a variety of parts and tasks. UWME.

    Devasia’s team has done simulations with autonomous vehicles that demonstrated the advantages of using the new delayed self-reinforcement algorithm. The researchers found that more than triple the number of cars can move through the traffic light when compared to human drivers. Even with a 50-50 mix of autonomous vehicles and human-driven cars, there is a 30% increase in the number of cars that can get through the green light with the algorithm.

    The lab is also interested in how multiple robots can be programmed to move at the same time so that they can carry large objects. In manufacturing, if a group of robots moves at different speeds or directions, it could damage the item they’re carrying.

    Students in Devasia’s lab also built robots to carry large objects such as composite parts and applied the algorithm to them. They found only a small deviation between when the first and last robot moved.

    Tiwari, who worked on the paper as a Ph.D. student, is grateful for his time in Devasia’s lab. This summer, he will become an assistant professor at IIT Gandhinagar India, where he plans to continue his research on networked systems and control.

    “Studying the propagation of information through networks has not only enhanced my understanding of natural flocking behavior, but also allowed me to work in promising engineering applications for control of robotic and vehicular networks,” he says.

    Devasia looks forward to applying the new algorithm to advanced manufacturing projects in the Boeing Advanced Research Center (BARC). He’s excited to continue to explore how one robot knows what the other is doing.

    “How information propagates is really important,” he says.


    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    Mechanical engineering is one of the broadest and oldest of the engineering disciplines and therefore provides some of the strongest interdisciplinary opportunities in the engineering profession. Power utilization (and power generation) is often used to describe the focus of mechanical engineering. Within this focus are such diverse topics as thermodynamics, heat transfer, fluid mechanics, machine design, mechanics of materials, manufacturing, stress analysis, system dynamics, numerical modeling, vibrations, turbomachinery, combustion, heating, ventilating, and air conditioning. Degrees in mechanical engineering open doors to careers not only in the engineering profession but also in business, law, medicine, finance, and other non-technical professions.

    About The University of Washington College of Engineering

    Mission, Facts, and Stats

    Our mission is to develop outstanding engineers and ideas that change the world.

    275 faculty (25.2% women)

    128 NSF Young Investigator/Early Career Awards since 1984
    32 Sloan Foundation Research Awards
    2 MacArthur Foundation Fellows (2007 and 2011)

    A national leader in educating engineers, each year the College turns out new discoveries, inventions and top-flight graduates, all contributing to the strength of our economy and the vitality of our community.

    Engineering innovation

    Engineers drive the innovation economy and are vital to solving society’s most challenging problems. The College of Engineering is a key part of a world-class research university in a thriving hub of aerospace, biotechnology, global health and information technology innovation. Over 50% of The University of Washington startups in FY18 came from the College of Engineering.

    Commitment to diversity and access

    The College of Engineering is committed to developing and supporting a diverse student body and faculty that reflect and elevate the populations we serve. We are a national leader in women in engineering; 25.5% of our faculty are women compared to 17.4% nationally. We offer a robust set of diversity programs for students and faculty.

    Research and commercialization

    The University of Washington is an engine of economic growth, today ranked third in the nation for the number of startups launched each year, with 65 companies having been started in the last five years alone by UW students and faculty, or with technology developed here. The College of Engineering is a key contributor to these innovations, and engineering faculty, students or technology are behind half of all UW startups. In FY19, UW received $1.58 billion in total research awards from federal and nonfederal sources.


    The University of Washington is one of the world’s preeminent public universities. Our impact on individuals, on our region, and on the world is profound — whether we are launching young people into a boundless future or confronting the grand challenges of our time through undaunted research and scholarship. Ranked number 10 in the world in Shanghai Jiao Tong University rankings and educating more than 54,000 students annually, our students and faculty work together to turn ideas into impact and in the process transform lives and our world. For more about our impact on the world, every day.

    So, what defines us —the students, faculty and community members at The University of Washington? Above all, it’s our belief in possibility and our unshakable optimism. It’s a connection to others, both near and far. It’s a hunger that pushes us to tackle challenges and pursue progress. It’s the conviction that together we can create a world of good. Join us on the journey.

    The University of Washington is a public research university in Seattle, Washington, United States. Founded in 1861, The University of Washington is one of the oldest universities on the West Coast; it was established in downtown Seattle approximately a decade after the city’s founding to aid its economic development. Today, The University of Washington’s 703-acre main Seattle campus is in the University District above the Montlake Cut, within the urban Puget Sound region of the Pacific Northwest. The university has additional campuses in Tacoma and Bothell. Overall, The University of Washington encompasses over 500 buildings and over 20 million gross square footage of space, including one of the largest library systems in the world with more than 26 university libraries, as well as the UW Tower, lecture halls, art centers, museums, laboratories, stadiums, and conference centers. The University of Washington offers bachelor’s, master’s, and doctoral degrees through 140 departments in various colleges and schools, sees a total student enrollment of roughly 46,000 annually, and functions on a quarter system.

    The University of Washington is a member of the Association of American Universities and is classified among “R1: Doctoral Universities – Very high research activity”. According to the National Science Foundation, UW spent $1.41 billion on research and development in 2018, ranking it 5th in the nation. As the flagship institution of the six public universities in Washington state, it is known for its medical, engineering and scientific research as well as its highly competitive computer science and engineering programs. Additionally, The University of Washington continues to benefit from its deep historic ties and major collaborations with numerous technology giants in the region, such as Amazon, Boeing, Nintendo, and particularly Microsoft. Paul G. Allen, Bill Gates and others spent significant time at Washington computer labs for a startup venture before founding Microsoft and other ventures. The University of Washington’s 22 varsity sports teams are also highly competitive, competing as the Huskies in the Pac-12 Conference of the NCAA Division I, representing the United States at the Olympic Games, and other major competitions.

    The University of Washington has been affiliated with many notable alumni and faculty, including 21 Nobel Prize laureates and numerous Pulitzer Prize winners, Fulbright Scholars, Rhodes Scholars and Marshall Scholars.

    In 1854, territorial governor Isaac Stevens recommended the establishment of a university in the Washington Territory. Prominent Seattle-area residents, including Methodist preacher Daniel Bagley, saw this as a chance to add to the city’s potential and prestige. Bagley learned of a law that allowed United States territories to sell land to raise money in support of public schools. At the time, Arthur A. Denny, one of the founders of Seattle and a member of the territorial legislature, aimed to increase the city’s importance by moving the territory’s capital from Olympia to Seattle. However, Bagley eventually convinced Denny that the establishment of a university would assist more in the development of Seattle’s economy. Two universities were initially chartered, but later the decision was repealed in favor of a single university in Lewis County provided that locally donated land was available. When no site emerged, Denny successfully petitioned the legislature to reconsider Seattle as a location in 1858.

    In 1861, scouting began for an appropriate 10 acres (4 ha) site in Seattle to serve as a new university campus. Arthur and Mary Denny donated eight acres, while fellow pioneers Edward Lander, and Charlie and Mary Terry, donated two acres on Denny’s Knoll in downtown Seattle. More specifically, this tract was bounded by 4th Avenue to the west, 6th Avenue to the east, Union Street to the north, and Seneca Streets to the south.

    John Pike, for whom Pike Street is named, was the university’s architect and builder. It was opened on November 4, 1861, as the Territorial University of Washington. The legislature passed articles incorporating the University, and establishing its Board of Regents in 1862. The school initially struggled, closing three times: in 1863 for low enrollment, and again in 1867 and 1876 due to funds shortage. The University of Washington awarded its first graduate Clara Antoinette McCarty Wilt in 1876, with a bachelor’s degree in science.

    19th century relocation

    By the time Washington state entered the Union in 1889, both Seattle and The University of Washington had grown substantially. The University of Washington’s total undergraduate enrollment increased from 30 to nearly 300 students, and the campus’s relative isolation in downtown Seattle faced encroaching development. A special legislative committee, headed by The University of Washington graduate Edmond Meany, was created to find a new campus to better serve the growing student population and faculty. The committee eventually selected a site on the northeast of downtown Seattle called Union Bay, which was the land of the Duwamish, and the legislature appropriated funds for its purchase and construction. In 1895, The University of Washington relocated to the new campus by moving into the newly built Denny Hall. The University of Washington Regents tried and failed to sell the old campus, eventually settling with leasing the area. This would later become one of the University’s most valuable pieces of real estate in modern-day Seattle, generating millions in annual revenue with what is now called the Metropolitan Tract. The original Territorial University building was torn down in 1908, and its former site now houses the Fairmont Olympic Hotel.

    The sole-surviving remnants of The University of Washington’s first building are four 24-foot (7.3 m), white, hand-fluted cedar, Ionic columns. They were salvaged by Edmond S. Meany, one of The University of Washington’s first graduates and former head of its history department. Meany and his colleague, Dean Herbert T. Condon, dubbed the columns as “Loyalty,” “Industry,” “Faith”, and “Efficiency”, or “LIFE.” The columns now stand in the Sylvan Grove Theater.

    20th century expansion

    Organizers of the 1909 Alaska-Yukon-Pacific Exposition eyed the still largely undeveloped campus as a prime setting for their world’s fair. They came to an agreement with The University of Washington ‘s Board of Regents that allowed them to use the campus grounds for the exposition, surrounding today’s Drumheller Fountain facing towards Mount Rainier. In exchange, organizers agreed Washington would take over the campus and its development after the fair’s conclusion. This arrangement led to a detailed site plan and several new buildings, prepared in part by John Charles Olmsted. The plan was later incorporated into the overall University of Washington campus master plan, permanently affecting the campus layout.

    Both World Wars brought the military to campus, with certain facilities temporarily lent to the federal government. In spite of this, subsequent post-war periods were times of dramatic growth for The University of Washington. The period between the wars saw a significant expansion of the upper campus. Construction of the Liberal Arts Quadrangle, known to students as “The Quad,” began in 1916 and continued to 1939. The University’s architectural centerpiece, Suzzallo Library, was built in 1926 and expanded in 1935.

    After World War II, further growth came with the G.I. Bill. Among the most important developments of this period was the opening of the School of Medicine in 1946, which is now consistently ranked as the top medical school in the United States. It would eventually lead to The University of Washington Medical Center, ranked by U.S. News and World Report as one of the top ten hospitals in the nation.

    In 1942, all persons of Japanese ancestry in the Seattle area were forced into inland internment camps as part of Executive Order 9066 following the attack on Pearl Harbor. During this difficult time, university president Lee Paul Sieg took an active and sympathetic leadership role in advocating for and facilitating the transfer of Japanese American students to universities and colleges away from the Pacific Coast to help them avoid the mass incarceration. Nevertheless, many Japanese American students and “soon-to-be” graduates were unable to transfer successfully in the short time window or receive diplomas before being incarcerated. It was only many years later that they would be recognized for their accomplishments during The University of Washington’s Long Journey Home ceremonial event that was held in May 2008.

    From 1958 to 1973, The University of Washington saw a tremendous growth in student enrollment, its faculties and operating budget, and also its prestige under the leadership of Charles Odegaard. The University of Washington student enrollment had more than doubled to 34,000 as the baby boom generation came of age. However, this era was also marked by high levels of student activism, as was the case at many American universities. Much of the unrest focused around civil rights and opposition to the Vietnam War. In response to anti-Vietnam War protests by the late 1960s, the University Safety and Security Division became The University of Washington Police Department.

    Odegaard instituted a vision of building a “community of scholars”, convincing the Washington State legislatures to increase investment in The University of Washington. Washington senators, such as Henry M. Jackson and Warren G. Magnuson, also used their political clout to gather research funds for the University of Washington. The results included an increase in the operating budget from $37 million in 1958 to over $400 million in 1973, solidifying The University of Washington as a top recipient of federal research funds in the United States. The establishment of technology giants such as Microsoft, Boeing and Amazon in the local area also proved to be highly influential in the University of Washington’s fortunes, not only improving graduate prospects but also helping to attract millions of dollars in university and research funding through its distinguished faculty and extensive alumni network.

    21st century

    In 1990, The University of Washington opened its additional campuses in Bothell and Tacoma. Although originally intended for students who have already completed two years of higher education, both schools have since become four-year universities with the authority to grant degrees. The first freshman classes at these campuses started in fall 2006. Today both Bothell and Tacoma also offer a selection of master’s degree programs.

    In 2012, The University of Washington began exploring plans and governmental approval to expand the main Seattle campus, including significant increases in student housing, teaching facilities for the growing student body and faculty, as well as expanded public transit options. The University of Washington light rail station was completed in March 2015, connecting Seattle’s Capitol Hill neighborhood to The University of Washington Husky Stadium within five minutes of rail travel time. It offers a previously unavailable option of transportation into and out of the campus, designed specifically to reduce dependence on private vehicles, bicycles and local King County buses.

    The University of Washington has been listed as a “Public Ivy” in Greene’s Guides since 2001, and is an elected member of the American Association of Universities. Among the faculty by 2012, there have been 151 members of American Association for the Advancement of Science, 68 members of the National Academy of Sciences(US), 67 members of the American Academy of Arts and Sciences, 53 members of the National Academy of Medicine, 29 winners of the Presidential Early Career Award for Scientists and Engineers, 21 members of the National Academy of Engineering, 15 Howard Hughes Medical Institute Investigators, 15 MacArthur Fellows, 9 winners of the Gairdner Foundation International Award, 5 winners of the National Medal of Science, 7 Nobel Prize laureates, 5 winners of Albert Lasker Award for Clinical Medical Research, 4 members of the American Philosophical Society, 2 winners of the National Book Award, 2 winners of the National Medal of Arts, 2 Pulitzer Prize winners, 1 winner of the Fields Medal, and 1 member of the National Academy of Public Administration. Among The University of Washington students by 2012, there were 136 Fulbright Scholars, 35 Rhodes Scholars, 7 Marshall Scholars and 4 Gates Cambridge Scholars. UW is recognized as a top producer of Fulbright Scholars, ranking 2nd in the US in 2017.

    The Academic Ranking of World Universities has consistently ranked The University of Washington as one of the top 20 universities worldwide every year since its first release. In 2019, The University of Washington ranked 14th worldwide out of 500 by the ARWU, 26th worldwide out of 981 in the Times Higher Education World University Rankings, and 28th worldwide out of 101 in the Times World Reputation Rankings. Meanwhile, QS World University Rankings ranked it 68th worldwide, out of over 900.

    U.S. News & World Report ranked The University of Washington 8th out of nearly 1,500 universities worldwide for 2021, with The University of Washington’s undergraduate program tied for 58th among 389 national universities in the U.S. and tied for 19th among 209 public universities.

    In 2019, it ranked 10th among the universities around the world by SCImago Institutions Rankings. In 2017, the Leiden Ranking, which focuses on science and the impact of scientific publications among the world’s 500 major universities, ranked The University of Washington 12th globally and 5th in the U.S.

    In 2019, Kiplinger Magazine’s review of “top college values” named University of Washington 5th for in-state students and 10th for out-of-state students among U.S. public colleges, and 84th overall out of 500 schools. In the Washington Monthly National University Rankings The University of Washington was ranked 15th domestically in 2018, based on its contribution to the public good as measured by social mobility, research, and promoting public service.

  • richardmitnick 1:21 pm on April 21, 2023 Permalink | Reply
    Tags: "A New Kind of Symmetry Shakes Up Physics", , , Disparate observations physicists had made in the past 40 years were really manifestations of the same lurking symmetry., Mathematics, Physicists and mathematicians are collaborating to work out the mathematics of these new symmetries., , , The “higher symmetries” work like a one-way street-a notable contrast to all other symmetries in physics., The most important symmetries of 20th-century physics could be extended more broadly to apply in quantum field theory-the basic theoretical framework in which physicists work today.   

    From “Quanta Magazine” : “A New Kind of Symmetry Shakes Up Physics” 

    From “Quanta Magazine”

    Kevin Hartnett

    The symmetries of 20th century physics were built on points. Higher symmetries are based on one-dimensional lines. Credit: Samuel Velasco/Quanta Magazine

    It is not an exaggeration to say that every major advance in physics for more than a century has turned on revelations about symmetry. It’s there at the dawn of General Relativity, in the birth of the Standard Model, in the hunt for the Higgs.

    For that reason, research across physics is now building to a crescendo. It was touched off by a 2014 paper which demonstrated that the most important symmetries of 20th-century physics could be extended more broadly to apply in quantum field theory, the basic theoretical framework in which physicists work today.

    This reformulation, which crystallized earlier work in the area, revealed that disparate observations physicists had made in the past 40 years were really manifestations of the same lurking symmetry. In doing so, it created an organizing principle that physicists could use to categorize and understand phenomena. “That’s really a stroke of genius,” said Nathaniel Craig, a physicist at the University of California-Santa Barbara.

    The principle identified in the paper came to be known as “higher symmetries.” The name reflects the way the symmetries apply to higher-dimensional objects such as lines, rather than lower-dimensional objects such as particles at single points in space. By giving the symmetry a name and language and by identifying places it had been observed before, the paper prompted physicists to search for other places it might appear.

    Physicists and mathematicians are collaborating to work out the mathematics of these new symmetries — and in some cases they’re discovering that the symmetries work like a one-way street, a notable contrast to all other symmetries in physics. At the same time, physicists are applying the symmetries to explain a wide range of questions, from the decay rate of certain particles to novel phase transitions like the fractional quantum Hall effect.

    “By putting a different perspective on a known sort of physical problem, it just opened up a huge new area,” said Sakura Schafer-Nameki, a physicist at the University of Oxford (UK).

    Symmetry Matters

    To understand why a paper that merely points out the breadth of lurking symmetries can make such a big impact, it helps to first understand how symmetry makes life easier for physicists. Symmetry means fewer details to keep track of. That’s true whether you’re doing high-energy physics or laying bathroom tile.

    The symmetries of a bathroom tile are spatial symmetries — each can be rotated, flipped upside down or moved to a new spot. Spatial symmetries play an important simplifying role in physics too. They’re prominent in Einstein’s theory of space-time — and the fact that they pertain to our universe means physicists have one less thing to worry about.

    “If you’re doing an experiment in a lab and you rotate it, that shouldn’t change your answer,” said Nathan Seiberg, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey.

    Nathan Seiberg was a co-author on the 2014 paper that developed the notion of higher symmetries. Credit: Andrea Kane/Institute for Advanced Study.

    The symmetries that are most important in physics today are subtler than spatial symmetries, but they carry the same meaning: They’re constraints on the ways that you can transform something to ensure that it’s still the same.

    In an epochal insight in 1915, the mathematician Emmy Noether formalized the relationship between symmetries and conservation laws.

    Mathematician Emmy Noether. Symmetry.

    For example, symmetries in time — it doesn’t matter if you run your experiment today or tomorrow — mathematically imply the law of conservation of energy. Rotational symmetries lead to the law of conservation of angular momentum.

    “Every conservation law is associated with a symmetry, and every symmetry is associated with a conservation law,” Seiberg said. “It’s well understood and it’s very deep.”

    This is just one of the ways that symmetries help physicists understand the universe.

    Physicists would like to create a taxonomy of physical systems, classifying like with like, in order to know when insights from one can be applied to another. Symmetries are a good organizing principle: All systems exhibiting the same symmetry go in the same bucket.

    Furthermore, if physicists know a system possesses a given symmetry, they can avoid a lot of the mathematical work of describing how it behaves. The symmetries constrain the possible states of the system, which means they limit the potential answers to the complicated equations that characterize the system.

    “Typically, some random physical equations are unsolvable, but if you have enough symmetry, then the symmetry constrains the possible answers. You can say the solution must be this because it’s the only symmetric thing,” said Theo Johnson-Freyd of the Perimeter Institute for Theoretical Physics in Waterloo, Canada.

    Symmetries convey elegance, and their presence can be obvious in hindsight. But until physicists identify their influence, related phenomena can remain distinct. Which is what happened with a host of observations physicists made starting in the early 1970s.

    Fields and Strings

    The conservation laws and symmetries of 20th-century physics take pointlike particles as their primary objects. But in modern quantum field theories, quantum fields are the most basic objects, and particles are just fluctuations in these fields. And within these theories it’s often necessary to go beyond points and particles to think about one-dimensional lines, or strings (which are conceptually distinct from the strings in string theory).

    In 1973, physicists described [Nuclear Physics B (below)] an experiment that involved placing a superconducting material between poles of a magnet. They observed that as they increased the strength of the magnetic field, particles arranged themselves along one-dimensional superconducting threads running between the magnetic poles.

    The next year Kenneth Wilson identified strings — Wilson lines [Physical Review D (below)]— in the setting of classical electromagnetism. Strings also appear in the way the strong force acts among quarks, which are the elementary particles that make up a proton. Separate a quark from its antiquark, and a string forms between them that pulls them back together.

    The point is that strings play an important role in many areas of physics. At the same time, they’re mismatched to traditional conservation laws and symmetries, which are expressed in terms of particles.

    “The modern thing is to say we’re not only interested in the properties of points; we’re interested in the properties of lines or strings, and there can also be conservation laws for them,” said Seiberg, who co-wrote the 2014 paper along with Davide Gaiotto of the Perimeter Institute, Anton Kapustin of the California Institute of Technology, and Brian Willett, who was at the time a postdoc at the Institute for Advanced Study.

    The paper presented a way of measuring charge along a string and establishing that charge remains conserved as the system evolves, just as total charge is always conserved for particles. And the team did it by shifting their attention from the string itself.

    Seiberg and his colleagues imagined the one-dimensional string as being surrounded by a surface, a two-dimensional plane, so that it looked like a line drawn on a sheet of paper. Instead of measuring charge along the string, they described a method for measuring the total charge across the surface surrounding the string.

    “The really new thing is you emphasize the charged object, and you think about [surfaces] that surround it,” Schafer-Nameki said.

    The four authors then considered what happens to the surrounding surface as the system evolves. Maybe it warps or twists or otherwise changes from the completely flat surface they measured originally. Then they demonstrated that even as the surface deforms, the total charge along it remains the same.

    That is, if you measure charge at every point on a piece of paper, then distort the paper and measure again, you’ll get the same number. You can say that charge is conserved along the surface, and since the surface is indexed to the string, you can say it’s conserved along the string, too — regardless of what kind of string you started with.

    “The mechanics of a superconducting string and a strong-force string are completely different, yet the mathematics of these strings and the conservation [laws] are exactly the same,” Seiberg said. “That’s the beauty of this whole idea.”

    Equivalent Surfaces

    The suggestion that a surface remains the same — has the same charge — even after it’s deformed echoes concepts from the mathematical field of topology. In topology, mathematicians classify surfaces according to whether one can be deformed into the other without any ripping. According to this viewpoint, a perfect sphere and a lopsided ball are equivalent, since you can inflate the ball to get the sphere. But a sphere and an inner tube are not, as you’d have to gash the sphere to get the inner tube.

    Similar thinking about equivalence applies to surfaces around strings — and by extension, the quantum field theories inside of which those surfaces are drawn, Seiberg and his co-authors wrote. They referred to their method of measuring charge on surfaces as a topological operator. The word “topological” conveys that sense of overlooking insignificant variations between a flat surface and a warped one. If you measure the charge on each, and it comes out the same, you know that the two systems can be smoothly deformed into each other.

    Topology allows mathematicians to look past minor variations to focus on fundamental ways in which different shapes are the same. Similarly, higher symmetries provide physicists with a new way of indexing quantum systems, the authors concluded. Those systems may look completely different from each other, but in a deep way they might really obey the same rules. Higher symmetries can detect that — and by detecting it, they allow physicists to take knowledge about better-understood quantum systems and apply it to others.

    “The development of all these symmetries is like developing a series of ID numbers for a quantum system,” said Shu-Heng Shao, a theoretical physicist at Stony Brook University. “Sometimes two seemingly unrelated quantum systems turn out to have the same set of symmetries, which suggests they might be the same quantum system.”

    Despite these elegant insights about strings and symmetries in quantum field theories, the 2014 paper didn’t spell out any dramatic ways of applying them. Equipped with new symmetries, physicists might hope to be able to answer new questions — but at the time, higher symmetries were only immediately useful for re-characterizing things physicists already knew. Seiberg recalls being disappointed that they couldn’t do more than that.

    “I remember going around thinking, ‘We need a killer app,’” he said.

    From New Symmetries to New Mathematics

    To write a killer app, you need a good programming language. In physics, mathematics is that language, explaining in a formal, rigorous way how symmetries work together. Following the landmark paper, mathematicians and physicists started by investigating how higher symmetries could be expressed in terms of objects called groups, which are the main mathematical structure used to describe symmetries.

    A group encodes all the ways the symmetries of a shape or a system can be combined. It establishes the rules for how the symmetries operate and tells you what positions the system can end up in following symmetry transformations (and which positions, or states, can never occur).

    Group encoding work is expressed in the language of algebra. In the same way that order matters when you’re solving an algebraic equation (dividing 4 by 2 is not the same as dividing 2 by 4), the algebraic structure of a group reveals how order matters when you’re applying symmetry transformations, including rotations.

    “Understanding algebraic relationships between transformations is a precursor to any application,” said Clay Córdova of the University of Chicago. “You can’t understand how the world is constrained by rotations until you understand ‘What are rotations?’”

    Merrill Sherman/Quanta Magazine.

    By investigating those relationships, two separate teams — one involving Córdova and Shao and one that includes researchers at Stony Brook and the University of Tokyo — discovered that even in realistic quantum systems, there are non-invertible symmetries that fail to conform to the group structure, a feature that every other important type of symmetry in physics fits into. Instead, these symmetries are described by related objects called categories which have more relaxed rules for how symmetries can be combined.

    For example, in a group, every symmetry is required to have an inverse symmetry — an operation that undoes it and sends the object it acts on back to where it started. But in separate papers published last year, the two groups showed that some higher symmetries are non-invertible, meaning once you apply them to a system, you can’t get back to where you started.

    This non-invertibility reflects the way that a higher symmetry can transform a quantum system into a superposition of states, in which it is probabilistically two things at once. From there, there’s no road back to the original system. To capture this more complicated way higher symmetries and non-invertible symmetries interact, researchers including Johnson-Freyd have developed a new mathematical object called a higher fusion category.

    “It’s the mathematical edifice that describes the fusions and interactions of all these symmetries,” Córdova said. “It tells you all the algebraic possibilities for how they can interact.”

    Higher fusion categories help to define the non-invertible symmetries that are mathematically possible, but they don’t tell you which symmetries are useful in specific physical situations. They establish the parameters of a hunt on which physicists then embark.

    “As a physicist the exciting thing is the physics we get out of it. It shouldn’t just be math for the sake of math,” Schafer-Nameki said.

    Early Applications

    Equipped with higher symmetries, physicists are also reevaluating old cases in light of new evidence.

    For example, in the 1960s physicists noticed a discrepancy in the decay rate of a particle called the pion. Theoretical calculations said it should be one thing, experimental observations said another. In 1969, two papers seemed to resolve the tension by showing that the quantum field theory which governs pion decay does not actually possess a symmetry that physicists thought it did. Without that symmetry, the discrepancy disappeared.

    But last May, three physicists proved that the 1969 verdict was only half the story. It wasn’t just that the presupposed symmetry wasn’t there — it was that higher symmetries were. And when those symmetries were incorporated into the theoretical picture, the predicted and observed decay rates matched exactly.

    “We can reinterpret this mystery of the pion decay not in terms of the absence of symmetry but in terms of the presence of a new kind of symmetry,” said Shao, a co-author of the paper.

    Similar reexamination has taken place in condensed matter physics. Phase transitions occur when a physical system switches from one state of matter to another. At a formal level, physicists describe those changes in terms of symmetries being broken: Symmetries that pertained in one phase no longer apply in the next.

    But not all phases have been neatly described by symmetry-breaking. One, called the fractional quantum Hall effect, involves the spontaneous reorganization of electrons, but without any apparent symmetry being broken. This made it an uncomfortable outlier within the theory of phase transitions. That is, until a paper in 2018 by Xiao-Gang Wen of the Massachusetts Institute of Technology helped establish that the quantum Hall effect does in fact break a symmetry — just not a traditional one.

    “You can think of [it] as symmetry-breaking if you generalize your notion of symmetry,” said Ashvin Vishwinath of Harvard University.

    These early applications of higher and non-invertible symmetries — to the pion decay rate, and to the understanding of the fractional quantum Hall effect — are modest compared to what physicists anticipate.

    In condensed matter physics, researchers hope that higher and non-invertible symmetries will help them with the fundamental task of identifying and classifying all possible phases of matter. And in particle physics, researchers are looking to higher symmetries to assist with one of the biggest open questions of all: what principles organize physics beyond the Standard Model.

    “I want to get the Standard Model out of a consistent theory of quantum gravity, and these symmetries play a critical role,” said Mirjam Cvetic of the University of Pennsylvania.

    It will take a while to fully reorient physics around an expanded understanding of symmetry and a broader notion of what makes systems the same. That so many physicists and mathematicians are joining in the effort suggests they think it will be worth it.

    “I have not yet seen shocking results that we didn’t know before, but I have no doubt it’s quite likely this will happen, because this is clearly a much better way of thinking about the problem,” Seiberg said.

    Nuclear Physics B 1973
    Physical Review D 1974

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    Formerly known as Simons Science News, Quanta Magazine is an editorially independent online publication launched by the Simons Foundation to enhance public understanding of science. Why Quanta? Albert Einstein called photons “quanta of light.” Our goal is to “illuminate science.” At Quanta Magazine, scientific accuracy is every bit as important as telling a good story. All of our articles are meticulously researched, reported, edited, copy-edited and fact-checked.

  • richardmitnick 12:39 pm on March 4, 2023 Permalink | Reply
    Tags: "Stick to your lane - hidden order in chaotic crowds", Mathematical research brings new understanding of crowd formation and behaviour., Mathematics,   

    From The University of Bath (UK) : “Stick to your lane – hidden order in chaotic crowds” 

    From The University of Bath (UK)

    Vittoria D’Alessio
    +44 (0)1225 383135

    Mathematical research brings new understanding of crowd formation and behaviour.

    Tilted lanes formed by two groups of people moving in opposite directions. The inclination results from a “pass on the right” traffic rule. Credit: University of Bath.

    Have you ever wondered how pedestrians ‘know’ to fall into lanes when they are moving through a crowd, without the matter being discussed or even given conscious thought?

    A new theory developed by mathematicians at the University of Bath led by Professor Tim Rogers explains this phenomenon, and is able to predict when lanes will be curved as well as straight. The theory can even describe the tilt of a wonky lane when people are in the habit of passing on one side rather than the other (for instance, in a situation where they are often reminded to ‘pass on the right’).

    This mathematical analysis unifies conflicting viewpoints on the origin of lane formation, and it reveals a new class of structures that in daily life may go unnoticed. The discovery, reported this week in the prestigious journal Science [below], constitutes a major advance in the interdisciplinary science of ‘active matter’ – the study of group behaviours in interacting populations ranging in scale from bacteria to herds of animals.

    Tested in arenas

    To test their theory, the researchers asked a group of volunteers to walk across an experimental arena that mimicked different layouts, with changes to entrance and exit gates.

    One arena was set up in the style of King’s Cross Station in London. When the researchers looked at the video footage from the experiment, they observed mathematical patterns taking shape in real life.

    Professor Rogers said: “At a glance, a crowd of pedestrians attempting to pass through two gates might seem disorderly but when you look more closely, you see the hidden structure. Depending on the layout of the space, you may observe either the classic straight lanes or more complex curved patterns such as ellipses, parabolas, and hyperbolas”.

    Lane formation

    The single-file processions formed at busy zebra crossings are only one example of lane formation, and this study is likely to have implications for a range of scientific disciplines, particularly in the fields of physics and biology. Similar structures can also be formed by inanimate molecules, such as charged particles or organelles in a cell.

    Until now, scientists have given several different explanations for why human crowds and other active systems naturally self-organize into lanes, but none of these theories have been verified. The Bath team used a new analytical approach, inspired by Albert Einstein’s theory of Brownian motion, which makes predictions that can be tested.

    Encouraged by the way their theory agreed with the numerical simulations of colliding particles, they then teamed up with Professor Bogdan Bacik – an experimentalist from the Academy of Physical Education in Katowice, Poland – and ran a series of experiments (such as the one modelled on King’s Cross) using human crowds.

    Lead author Dr Karol Bacik said: “Lane formation doesn’t require conscious thought – the participants of the experiment were not aware that they had arranged themselves into well-defined mathematical curves.

    “The order emerges spontaneously when two groups with different objectives cross paths in a crowded space and try to avoid crashing into each other. The cumulative effect of lots of individual decisions inadvertently results in lanes forming.”

    The researchers also tested the effects of externally imposed traffic rules – namely, they instructed the participants to pass others on the right. In agreement with the theoretical prediction, adding this rule changed the lane structure.

    “When pedestrians have a preference for right turns, the lanes end up tilting and this introduces frustration that slows people down,” said Dr Bacik.

    “What we’ve developed is a neat mathematical theory that forecasts the propensity for lane formation in any given system,” said Professor Rogers, adding: “We now know that much more structure exists than previously thought.”


    Fig. 1. Lane formation examples and kinetic model.
    (A and B) Lane formation examples. Bidirectional pedestrian flow realized in a controlled experiment is shown in (A). An agent-based simulation of driven hard spheres is shown in (B). (C to E) Kinetic model. In (C), a setup sketch focusing on a “+”-type agent moving with a preferred velocity vey (gray circle) is shown. The presence of the “−”-type agent (black circle) moving with velocity −vey alters its trajectory, so instead of advancing by vΔtey, its displacement within a Δt time interval is given by vΔtey + G(x), where G is the collisional operator and x is the initial lateral offset between the two agents. In (D), the x component of the collisional displacement extracted from the pedestrian experiment [see (43) for inference procedure] is shown. Each dark gray circle corresponds to one interaction event, the black line is a running average of these data, and the light gray shading shows the corresponding standard deviation. In (E), a diagram explaining the hydrodynamic Eq. 3 is shown. In the linearized model, the evolution of agent density can be understood as a superposition of five processes: active migration in the preferred direction, density- and inhomogeneity-induced drift (which, in a symmetric system, act in orthogonal directions), and homogeneous and inhomogeneous diffusion.

    Fig. 2. Stability analysis
    (A) The growth rate of lane-like Fourier modes at the moment of maximal linear growth t* extracted through a Fourier transform from an ensemble of 5000 simulations of 300 hard spheres moving in a doubly periodic domain with preferred velocities subtending angle ψ (large angles correspond to black squares, as explained by the color bar at the top right), as well as hard ellipses with varying aspect ratio η (circles of varying intensity; see inset). The empirically measured growth rate shows an agreement with the theoretical dispersion relation with a global maximum for λmax ≈ 2D. The simulation details can be found in section IV of (43). (B) The maximal growth rate σmax can also be approximated using a simple heuristic scaling σmax ∝ vρ0μ.

    For further illustrations see the science paper.

    See the full article here.

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    The University of Bath (UK) is a public research university located in Bath, Somerset, United Kingdom. It received its royal charter in 1966, along with a number of other institutions following the Robbins Report. Like the University of Bristol (UK) and University of the West of England-Bristol (UK), Bath can trace its roots to the Merchant Venturers’ Technical College, established in Bristol as a school in 1595 by the Society of Merchant Venturers. The university’s main campus is located on Claverton Down, a site overlooking the city of Bath, and was purpose-built, constructed from 1964 in the modernist style of the time.

    In the 2014 Research Excellence Framework, 32% of Bath’s submitted research activity achieved the highest possible classification of 4*, defined as world-leading in terms of originality, significance and rigour. 87% was graded 4*/3*, defined as world-leading/internationally excellent. The annual income of the institution for 2017–18 was £287.9 million of which £37.0 million was from research grants and contracts, with an expenditure of £283.1 million.

    The university is a member of the Association of Commonwealth Universities (UK), the Association of MBAs, the European Quality Improvement System, the European University Association (EU), Universities UK and GW4 (UK).

  • richardmitnick 3:19 pm on February 25, 2023 Permalink | Reply
    Tags: "Theory can sort order from chaos in complex quantum systems", A water droplet gleaming on a leaf may look motionless but inside over a sextillion molecules are vibrating nonstop., Another direction that is being pursued at Rice where this theory can help is the problem of making a quantum computer that behaves as much as possible in a clocklike way., , Aside from their nonstop vibrations-which happen without light-electrons can have quantum-level interactions that sometimes lead to a more dramatic turn., , Collaborators at the University of Illinois Urbana-Champaign showed that the theory can predict the nature of the motions in a chlorophyll molecule when it absorbs energy from sunlight., Development could spark advances in computing and electrochemical and biological systems., Hydrogen and oxygen atoms and the subatomic particles within them-the nuclei and electrons-constantly move and interact., Mathematics, Nothing is ever completely still on the molecular level especially when quantum physics plays a role., , , , The theory applies to any sufficiently complex quantum system and may give insights into building better quantum computers., The theory developed by Rice’s Peter Wolynes and Oxford’s David Logan gives a simple prediction for the threshold at which large quantum systems switch from orderly motion to random erratic motion, , Things are more complicated when the molecules also dramatically change structure-for instance as a result of a chemical reaction., When Wolynes and Logan first posed the question of predicting the regularity or randomness of quantum motion they tested their math against observations of vibrational motions in individual molecules.   

    From Rice University And From The University of Oxford (UK): “Theory can sort order from chaos in complex quantum systems” 

    From Rice University


    U Oxford bloc

    From The University of Oxford (UK)

    Silvia Cernea Clark

    Development could spark advances in computing and electrochemical and biological systems.

    It’s not easy to make sense of quantum-scale motion, but a new mathematical theory developed by scientists at Rice University and Oxford University could help — and may provide insight into improving a variety of computing, electrochemical and biological systems.

    Credit: CC0 Public Domain.

    The theory developed by Rice theorist Peter Wolynes and Oxford theoretical chemist David Logan gives a simple prediction for the threshold at which large quantum systems switch from orderly motion like a clock to random, erratic motion like asteroids moving around in the early solar system. Using a computational analysis of a photosynthesis model, collaborators at the University of Illinois Urbana-Champaign showed that the theory can predict the nature of the motions in a chlorophyll molecule when it absorbs energy from sunlight.

    Peter Wolynes is Rice’s Bullard-Welch Foundation Professor of Science and a professor of chemistry, of biochemistry and cell biology, of physics and astronomy and of materials science and nanoengineering and co-director of the Center for Theoretical Biological Physics at Rice. (Photo by Gustavo Raskosky/Rice University)

    David Logan is the Coulson Professor of Theoretical Chemistry at the University of Oxford. (Photo courtesy of David Logan)

    The theory applies to any sufficiently complex quantum system and may give insights into building better quantum computers. It could also, for instance, help design features of next-generation solar cells or perhaps make batteries last longer.

    The study is published this week in the PNAS [below].

    Nothing is ever completely still on the molecular level especially when quantum physics plays a role. A water droplet gleaming on a leaf may look motionless but inside over a sextillion molecules are vibrating nonstop. Hydrogen and oxygen atoms and the subatomic particles within them — the nuclei and electrons — constantly move and interact.

    “In thinking about the motions of individual molecules at quantum scale, there is often this comparison to the way we think of the solar system,” Wolynes said. “You learn that there are eight planets in our solar system, each one with a well-defined orbit. But in fact, the orbits interact with each other. Nevertheless, the orbits are very predictable. You can go to a planetarium, and they’ll show you what the sky looked like 2,000 years ago. A lot of the motions of the atoms in molecules are exactly that regular or clocklike.”

    When Wolynes and Logan first posed the question of predicting the regularity or randomness of quantum motion they tested their math against observations of vibrational motions in individual molecules.

    “You only have to know two things about a molecule to be able to analyze its quantum motion patterns,” Wolynes said. “First, you need to know the vibrational frequencies of its particles, that’s to say the frequencies at which the vibrations occur which are like the orbits, and, second, how these vibrations nonlinearly interact with each other. These anharmonic interactions depend mostly on the mass of atoms. For organic molecules, you can predict how strongly those vibrational orbits would interact with one other.”

    Things are more complicated when the molecules also dramatically change structure-for instance as a result of a chemical reaction.

    “As soon as we start looking at molecules that chemically react or rearrange their structure, we know that there’s at least some element of unpredictability or randomness in the process because, even in classical terms, the reaction either happens, or it doesn’t happen,” Wolynes said. “When we try to understand how chemical changes occur, there’s this question: Is the overall motion more clocklike or is it more irregular?”

    Aside from their nonstop vibrations, which happen without light, electrons can have quantum-level interactions that sometimes lead to a more dramatic turn.

    “Because they’re very light, electrons normally move thousands of times faster than the centers of the atoms, the nuclei,” he said. “So though they are constantly moving, the electrons’ orbits smoothly adjust to what the nuclei do. But every now and again, the nuclei come to a place where the electronic energies will almost be equal whether the excitation is on one molecule or on the other. That’s what’s called a surface crossing. At that point, the excitation has a chance to jump from one electronic level to another.”

    Predicting at which point the transfer of energy that takes place during photosynthesis turns from orderly motion to randomness or dissipation would take a significant amount of time and effort by direct computation.

    Logan-Wolynes theory: predicting energy transfer in a chlorophyll model.

    “It is very nice that we have a very simple formula that determines when this happens,” said Martin Gruebele, a chemist at the University of Illinois Urbana-Champaign and co-author on the study who is a part of the joint Rice-Illinois Center for Adapting Flaws into Features (CAFF) funded by the National Science Foundation. “That’s something we just didn’t have before and figuring it out required very lengthy calculations.”

    The Logan-Wolynes theory opens up a wide array of scientific inquiry ranging from the theoretical exploration of the fundamentals of quantum mechanics to practical applications.

    “The Logan-Wolynes theory did pretty well in terms of telling you at roughly what energy input you’d get a change in quantum-system behavior,” Wolynes said. “But one of the interesting things that the large-scale computations of (co-author Chenghao) Zhang and Gruebele found is that there are these exceptions that stand out from all the possible orbiting patterns you might have. Occasionally there’s a few stragglers where simple motions persist for long times and don’t seem to get randomized. One of the questions we’re going to pursue in the future is how much that persistent regularity is actually influencing processes like photosynthesis.

    “Another direction that is being pursued at Rice where this theory can help is the problem of making a quantum computer that behaves as much as possible in a clocklike way,” he said. “You don’t want your computers to be randomly changing information. The larger and more sophisticated you make a computer, the likelier it is that you’ll run into some kind of randomization effects.”

    Gruebele and collaborators at Illinois also plan to use these ideas in other scientific contexts. “One of our goals, for instance, is to design better human-built light-harvesting molecules that might consist of carbon dots that can transfer the energy to their periphery where it can be harvested,” Gruebele said.

    Wolynes is Rice’s Bullard-Welch Foundation Professor of Science and a professor of chemistry, of biochemistry and cell biology, of physics and astronomy and of materials science and nanoengineering and co-director of its Center for Theoretical Biological Physics (CTBP), which is funded by the National Science Foundation. Logan is the Coulson Professor of Theoretical Chemistry at Oxford. Gruebele is the James R. Eiszner Endowed Chair in Chemistry and Zhang is a graduate student in physics at the University of Illinois Urbana-Champaign.

    The James R. Eiszner Chair in Chemistry and the Physics Department at Illinois, the Bullard-Welch Chair at Rice (C-0016) and the National Science Foundation (PHY-2019745) supported the research.


    From the science paper:


    See the full article here .


    Stem Education Coalition

    U Oxford campus

    The University of Oxford

    Universitas Oxoniensis

    The University of Oxford [a.k.a. The Chancellor, Masters and Scholars of the University of Oxford] is a collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the world’s second-oldest university in continuous operation. It grew rapidly from 1167 when Henry II banned English students from attending the University of Paris [Université de Paris](FR). After disputes between students and Oxford townsfolk in 1209, some academics fled north-east to Cambridge where they established what became the The University of Cambridge (UK). The two English ancient universities share many common features and are jointly referred to as Oxbridge.

    The university is made up of thirty-nine semi-autonomous constituent colleges, six permanent private halls, and a range of academic departments which are organised into four divisions. All the colleges are self-governing institutions within the university, each controlling its own membership and with its own internal structure and activities. All students are members of a college. It does not have a main campus, and its buildings and facilities are scattered throughout the city centre. Undergraduate teaching at Oxford consists of lectures, small-group tutorials at the colleges and halls, seminars, laboratory work and occasionally further tutorials provided by the central university faculties and departments. Postgraduate teaching is provided predominantly centrally.

    Oxford operates the world’s oldest university museum, as well as the largest university press in the world and the largest academic library system nationwide. In the fiscal year ending 31 July 2019, the university had a total income of £2.45 billion, of which £624.8 million was from research grants and contracts.

    Oxford has educated a wide range of notable alumni, including 28 prime ministers of the United Kingdom and many heads of state and government around the world. As of October 2020, 72 Nobel Prize laureates, 3 Fields Medalists, and 6 Turing Award winners have studied, worked, or held visiting fellowships at the University of Oxford, while its alumni have won 160 Olympic medals. Oxford is the home of numerous scholarships, including the Rhodes Scholarship, one of the oldest international graduate scholarship programmes.

    The University of Oxford’s foundation date is unknown. It is known that teaching at Oxford existed in some form as early as 1096, but it is unclear when a university came into being.

    It grew quickly from 1167 when English students returned from The University of Paris-Sorbonne [Université de Paris-Sorbonne](FR). The historian Gerald of Wales lectured to such scholars in 1188, and the first known foreign scholar, Emo of Friesland, arrived in 1190. The head of the university had the title of chancellor from at least 1201, and the masters were recognised as a universitas or corporation in 1231. The university was granted a royal charter in 1248 during the reign of King Henry III.

    The students associated together on the basis of geographical origins, into two ‘nations’, representing the North (northerners or Boreales, who included the English people from north of the River Trent and the Scots) and the South (southerners or Australes, who included English people from south of the Trent, the Irish and the Welsh). In later centuries, geographical origins continued to influence many students’ affiliations when membership of a college or hall became customary in Oxford. In addition, members of many religious orders, including Dominicans, Franciscans, Carmelites and Augustinians, settled in Oxford in the mid-13th century, gained influence and maintained houses or halls for students. At about the same time, private benefactors established colleges as self-contained scholarly communities. Among the earliest such founders were William of Durham, who in 1249 endowed University College, and John Balliol, father of a future King of Scots; Balliol College bears his name. Another founder, Walter de Merton, a Lord Chancellor of England and afterwards Bishop of Rochester, devised a series of regulations for college life. Merton College thereby became the model for such establishments at Oxford, as well as at the University of Cambridge. Thereafter, an increasing number of students lived in colleges rather than in halls and religious houses.

    In 1333–1334, an attempt by some dissatisfied Oxford scholars to found a new university at Stamford, Lincolnshire, was blocked by the universities of Oxford and Cambridge petitioning King Edward III. Thereafter, until the 1820s, no new universities were allowed to be founded in England, even in London; thus, Oxford and Cambridge had a duopoly, which was unusual in large western European countries.

    The new learning of the Renaissance greatly influenced Oxford from the late 15th century onwards. Among university scholars of the period were William Grocyn, who contributed to the revival of Greek language studies, and John Colet, the noted biblical scholar.

    With the English Reformation and the breaking of communion with the Roman Catholic Church, recusant scholars from Oxford fled to continental Europe, settling especially at the University of Douai. The method of teaching at Oxford was transformed from the medieval scholastic method to Renaissance education, although institutions associated with the university suffered losses of land and revenues. As a centre of learning and scholarship, Oxford’s reputation declined in the Age of Enlightenment; enrollments fell and teaching was neglected.

    In 1636, William Laud, the chancellor and Archbishop of Canterbury, codified the university’s statutes. These, to a large extent, remained its governing regulations until the mid-19th century. Laud was also responsible for the granting of a charter securing privileges for The University Press, and he made significant contributions to the Bodleian Library, the main library of the university. From the beginnings of the Church of England as the established church until 1866, membership of the church was a requirement to receive the BA degree from the university and “dissenters” were only permitted to receive the MA in 1871.

    The university was a centre of the Royalist party during the English Civil War (1642–1649), while the town favoured the opposing Parliamentarian cause. From the mid-18th century onwards, however, the university took little part in political conflicts.

    Wadham College, founded in 1610, was the undergraduate college of Sir Christopher Wren. Wren was part of a brilliant group of experimental scientists at Oxford in the 1650s, the Oxford Philosophical Club, which included Robert Boyle and Robert Hooke. This group held regular meetings at Wadham under the guidance of the college’s Warden, John Wilkins, and the group formed the nucleus that went on to found the Royal Society.

    Before reforms in the early 19th century, the curriculum at Oxford was notoriously narrow and impractical. Sir Spencer Walpole, a historian of contemporary Britain and a senior government official, had not attended any university. He said, “Few medical men, few solicitors, few persons intended for commerce or trade, ever dreamed of passing through a university career.” He quoted the Oxford University Commissioners in 1852 stating: “The education imparted at Oxford was not such as to conduce to the advancement in life of many persons, except those intended for the ministry.” Nevertheless, Walpole argued:

    “Among the many deficiencies attending a university education there was, however, one good thing about it, and that was the education which the undergraduates gave themselves. It was impossible to collect some thousand or twelve hundred of the best young men in England, to give them the opportunity of making acquaintance with one another, and full liberty to live their lives in their own way, without evolving in the best among them, some admirable qualities of loyalty, independence, and self-control. If the average undergraduate carried from university little or no learning, which was of any service to him, he carried from it a knowledge of men and respect for his fellows and himself, a reverence for the past, a code of honour for the present, which could not but be serviceable. He had enjoyed opportunities… of intercourse with men, some of whom were certain to rise to the highest places in the Senate, in the Church, or at the Bar. He might have mixed with them in his sports, in his studies, and perhaps in his debating society; and any associations which he had this formed had been useful to him at the time, and might be a source of satisfaction to him in after life.”

    Out of the students who matriculated in 1840, 65% were sons of professionals (34% were Anglican ministers). After graduation, 87% became professionals (59% as Anglican clergy). Out of the students who matriculated in 1870, 59% were sons of professionals (25% were Anglican ministers). After graduation, 87% became professionals (42% as Anglican clergy).

    M. C. Curthoys and H. S. Jones argue that the rise of organised sport was one of the most remarkable and distinctive features of the history of the universities of Oxford and Cambridge in the late 19th and early 20th centuries. It was carried over from the athleticism prevalent at the public schools such as Eton, Winchester, Shrewsbury, and Harrow.

    All students, regardless of their chosen area of study, were required to spend (at least) their first year preparing for a first-year examination that was heavily focused on classical languages. Science students found this particularly burdensome and supported a separate science degree with Greek language study removed from their required courses. This concept of a Bachelor of Science had been adopted at other European universities (The University of London (UK) had implemented it in 1860) but an 1880 proposal at Oxford to replace the classical requirement with a modern language (like German or French) was unsuccessful. After considerable internal wrangling over the structure of the arts curriculum, in 1886 the “natural science preliminary” was recognized as a qualifying part of the first-year examination.

    At the start of 1914, the university housed about 3,000 undergraduates and about 100 postgraduate students. During the First World War, many undergraduates and fellows joined the armed forces. By 1918 virtually all fellows were in uniform, and the student population in residence was reduced to 12 per cent of the pre-war total. The University Roll of Service records that, in total, 14,792 members of the university served in the war, with 2,716 (18.36%) killed. Not all the members of the university who served in the Great War were on the Allied side; there is a remarkable memorial to members of New College who served in the German armed forces, bearing the inscription, ‘In memory of the men of this college who coming from a foreign land entered into the inheritance of this place and returning fought and died for their country in the war 1914–1918’. During the war years the university buildings became hospitals, cadet schools and military training camps.


    Two parliamentary commissions in 1852 issued recommendations for Oxford and Cambridge. Archibald Campbell Tait, former headmaster of Rugby School, was a key member of the Oxford Commission; he wanted Oxford to follow the German and Scottish model in which the professorship was paramount. The commission’s report envisioned a centralised university run predominantly by professors and faculties, with a much stronger emphasis on research. The professional staff should be strengthened and better paid. For students, restrictions on entry should be dropped, and more opportunities given to poorer families. It called for an enlargement of the curriculum, with honours to be awarded in many new fields. Undergraduate scholarships should be open to all Britons. Graduate fellowships should be opened up to all members of the university. It recommended that fellows be released from an obligation for ordination. Students were to be allowed to save money by boarding in the city, instead of in a college.

    The system of separate honour schools for different subjects began in 1802, with Mathematics and Literae Humaniores. Schools of “Natural Sciences” and “Law, and Modern History” were added in 1853. By 1872, the last of these had split into “Jurisprudence” and “Modern History”. Theology became the sixth honour school. In addition to these B.A. Honours degrees, the postgraduate Bachelor of Civil Law (B.C.L.) was, and still is, offered.

    The mid-19th century saw the impact of the Oxford Movement (1833–1845), led among others by the future Cardinal John Henry Newman. The influence of the reformed model of German universities reached Oxford via key scholars such as Edward Bouverie Pusey, Benjamin Jowett and Max Müller.

    Administrative reforms during the 19th century included the replacement of oral examinations with written entrance tests, greater tolerance for religious dissent, and the establishment of four women’s colleges. Privy Council decisions in the 20th century (e.g. the abolition of compulsory daily worship, dissociation of the Regius Professorship of Hebrew from clerical status, diversion of colleges’ theological bequests to other purposes) loosened the link with traditional belief and practice. Furthermore, although the university’s emphasis had historically been on classical knowledge, its curriculum expanded during the 19th century to include scientific and medical studies. Knowledge of Ancient Greek was required for admission until 1920, and Latin until 1960.

    The University of Oxford began to award doctorates for research in the first third of the 20th century. The first Oxford D.Phil. in mathematics was awarded in 1921.

    The mid-20th century saw many distinguished continental scholars, displaced by Nazism and communism, relocating to Oxford.

    The list of distinguished scholars at the University of Oxford is long and includes many who have made major contributions to politics, the sciences, medicine, and literature. As of October 2020, 72 Nobel laureates and more than 50 world leaders have been affiliated with the University of Oxford.

    To be a member of the university, all students, and most academic staff, must also be a member of a college or hall. There are thirty-nine colleges of the University of Oxford (including Reuben College, planned to admit students in 2021) and six permanent private halls (PPHs), each controlling its membership and with its own internal structure and activities. Not all colleges offer all courses, but they generally cover a broad range of subjects.

    The colleges are:

    All-Souls College
    Balliol College
    Brasenose College
    Christ Church College
    Corpus-Christi College
    Exeter College
    Green-Templeton College
    Harris-Manchester College
    Hertford College
    Jesus College
    Keble College
    Kellogg College
    Linacre College
    Lincoln College
    Magdalen College
    Mansfield College
    Merton College
    New College
    Nuffield College
    Oriel College
    Pembroke College
    Queens College
    Reuben College
    St-Anne’s College
    St-Antony’s College
    St-Catherines College
    St-Cross College
    St-Edmund-Hall College
    St-Hilda’s College
    St-Hughs College
    St-John’s College
    St-Peters College
    Somerville College
    Trinity College
    University College
    Wadham College
    Wolfson College
    Worcester College

    The permanent private halls were founded by different Christian denominations. One difference between a college and a PPH is that whereas colleges are governed by the fellows of the college, the governance of a PPH resides, at least in part, with the corresponding Christian denomination. The six current PPHs are:

    Campion Hall
    Regent’s Park College
    St Benet’s Hall
    St-Stephen’s Hall
    Wycliffe Hall

    The PPHs and colleges join as the Conference of Colleges, which represents the common concerns of the several colleges of the university, to discuss matters of shared interest and to act collectively when necessary, such as in dealings with the central university. The Conference of Colleges was established as a recommendation of the Franks Commission in 1965.

    Teaching members of the colleges (i.e. fellows and tutors) are collectively and familiarly known as dons, although the term is rarely used by the university itself. In addition to residential and dining facilities, the colleges provide social, cultural, and recreational activities for their members. Colleges have responsibility for admitting undergraduates and organizing their tuition; for graduates, this responsibility falls upon the departments. There is no common title for the heads of colleges: the titles used include Warden, Provost, Principal, President, Rector, Master and Dean.

    Oxford is regularly ranked within the top 5 universities in the world and is currently ranked first in the world in the Times Higher Education World University Rankings, as well as the Forbes’s World University Rankings. It held the number one position in The Times Good University Guide for eleven consecutive years, and the medical school has also maintained first place in the “Clinical, Pre-Clinical & Health” table of The Times Higher Education World University Rankings for the past seven consecutive years. In 2021, it ranked sixth among the universities around the world by SCImago Institutions Rankings. The Times Higher Education has also recognised Oxford as one of the world’s “six super brands” on its World Reputation Rankings, along with The University of California-Berkeley, The University of Cambridge (UK), Harvard University, The Massachusetts Institute of Technology, and Stanford University. The university is fifth worldwide on the US News ranking. Its Saïd Business School came 13th in the world in The Financial Times Global MBA Ranking.
    Oxford was ranked ninth in the world in 2015 by The Nature Index, which measures the largest contributors to papers published in 82 leading journals. It is ranked fifth best university worldwide and first in Britain for forming CEOs according to The Professional Ranking World Universities, and first in the UK for the quality of its graduates as chosen by the recruiters of the UK’s major companies.

    In the 2018 Complete University Guide, all 38 subjects offered by Oxford rank within the top 10 nationally meaning Oxford was one of only two multi-faculty universities (along with Cambridge) in the UK to have 100% of their subjects in the top 10. Computer Science, Medicine, Philosophy, Politics and Psychology were ranked first in the UK by the guide.

    According to The QS World University Rankings by Subject, the University of Oxford also ranks as number one in the world for four Humanities disciplines: English Language and Literature, Modern Languages, Geography, and History. It also ranks second globally for Anthropology, Archaeology, Law, Medicine, Politics & International Studies, and Psychology.

    Rice University [formally William Marsh Rice University] is a private research university in Houston, Texas. It is situated on a 300-acre campus near the Houston Museum District and is adjacent to the Texas Medical Center.
    Opened in 1912 after the murder of its namesake William Marsh Rice, Rice is a research university with an undergraduate focus. Its emphasis on education is demonstrated by a small student body and 6:1 student-faculty ratio. The university has a very high level of research activity. Rice is noted for its applied science programs in the fields of artificial heart research, structural chemical analysis, signal processing, space science, and nanotechnology. Rice has been a member of the Association of American Universities since 1985 and is classified among “R1: Doctoral Universities – Very high research activity”.
    The university is organized into eleven residential colleges and eight schools of academic study, including the Wiess School of Natural Sciences, the George R. Brown School of Engineering, the School of Social Sciences, School of Architecture, Shepherd School of Music and the School of Humanities. Rice’s undergraduate program offers more than fifty majors and two dozen minors, and allows a high level of flexibility in pursuing multiple degree programs. Additional graduate programs are offered through the Jesse H. Jones Graduate School of Business and the Susanne M. Glasscock School of Continuing Studies. Rice students are bound by the strict Honor Code, which is enforced by a student-run Honor Council.
    Rice competes in 14 NCAA Division I varsity sports and is a part of Conference USA, often competing with its cross-town rival the University of Houston. Intramural and club sports are offered in a wide variety of activities such as jiu jitsu, water polo, and crew.
    The university’s alumni include more than two dozen Marshall Scholars and a dozen Rhodes Scholars. Given the university’s close links to National Aeronautics Space Agency, it has produced a significant number of astronauts and space scientists. In business, Rice graduates include CEOs and founders of Fortune 500 companies; in politics, alumni include congressmen, cabinet secretaries, judges, and mayors. Two alumni have won the Nobel Prize.


    Rice University’s history began with the demise of Massachusetts businessman William Marsh Rice, who had made his fortune in real estate, railroad development and cotton trading in the state of Texas. In 1891, Rice decided to charter a free-tuition educational institute in Houston, bearing his name, to be created upon his death, earmarking most of his estate towards funding the project. Rice’s will specified the institution was to be “a competitive institution of the highest grade” and that only white students would be permitted to attend. On the morning of September 23, 1900, Rice, age 84, was found dead by his valet, Charles F. Jones, and was presumed to have died in his sleep. Shortly thereafter, a large check made out to Rice’s New York City lawyer, signed by the late Rice, aroused the suspicion of a bank teller, due to the misspelling of the recipient’s name. The lawyer, Albert T. Patrick, then announced that Rice had changed his will to leave the bulk of his fortune to Patrick, rather than to the creation of Rice’s educational institute. A subsequent investigation led by the District Attorney of New York resulted in the arrests of Patrick and of Rice’s butler and valet Charles F. Jones, who had been persuaded to administer chloroform to Rice while he slept. Rice’s friend and personal lawyer in Houston, Captain James A. Baker, aided in the discovery of what turned out to be a fake will with a forged signature. Jones was not prosecuted since he cooperated with the district attorney, and testified against Patrick. Patrick was found guilty of conspiring to steal Rice’s fortune and he was convicted of murder in 1901 (he was pardoned in 1912 due to conflicting medical testimony). Baker helped Rice’s estate direct the fortune, worth $4.6 million in 1904 ($131 million today), towards the founding of what was to be called the Rice Institute, later to become Rice University. The board took control of the assets on April 29 of that year.

    In 1907, the Board of Trustees selected the head of the Department of Mathematics and Astronomy at Princeton University, Edgar Odell Lovett, to head the Institute, which was still in the planning stages. He came recommended by Princeton University‘s president, Woodrow Wilson. In 1908, Lovett accepted the challenge, and was formally inaugurated as the Institute’s first president on October 12, 1912. Lovett undertook extensive research before formalizing plans for the new Institute, including visits to 78 institutions of higher learning across the world on a long tour between 1908 and 1909. Lovett was impressed by such things as the aesthetic beauty of the uniformity of the architecture at the University of Pennsylvania, a theme which was adopted by the Institute, as well as the residential college system at University of Cambridge (UK) in England, which was added to the Institute several decades later. Lovett called for the establishment of a university “of the highest grade,” “an institution of liberal and technical learning” devoted “quite as much to investigation as to instruction.” [We must] “keep the standards up and the numbers down,” declared Lovett. “The most distinguished teachers must take their part in undergraduate teaching, and their spirit should dominate it all.”
    Establishment and growth

    In 1911, the cornerstone was laid for the Institute’s first building, the Administration Building, now known as Lovett Hall in honor of the founding president. On September 23, 1912, the 12th anniversary of William Marsh Rice’s murder, the William Marsh Rice Institute for the Advancement of Letters, Science, and Art began course work with 59 enrolled students, who were known as the “59 immortals,” and about a dozen faculty. After 18 additional students joined later, Rice’s initial class numbered 77, 48 male and 29 female. Unusual for the time, Rice accepted coeducational admissions from its beginning, but on-campus housing would not become co-ed until 1957.

    Three weeks after opening, a spectacular international academic festival was held, bringing Rice to the attention of the entire academic world.

    Per William Marsh Rice’s will and Rice Institute’s initial charter, the students paid no tuition. Classes were difficult, however, and about half of Rice’s students had failed after the first 1912 term. At its first commencement ceremony, held on June 12, 1916, Rice awarded 35 bachelor’s degrees and one master’s degree. That year, the student body also voted to adopt the Honor System, which still exists today. Rice’s first doctorate was conferred in 1918 on mathematician Hubert Evelyn Bray.

    The Founder’s Memorial Statue, a bronze statue of a seated William Marsh Rice, holding the original plans for the campus, was dedicated in 1930, and installed in the central academic quad, facing Lovett Hall. The statue was crafted by John Angel. In 2020, Rice students petitioned the university to take down the statue due to the founder’s history as slave owner.

    During World War II, Rice Institute was one of 131 colleges and universities nationally that took part in the V-12 Navy College Training Program, which offered students a path to a Navy commission.

    The residential college system proposed by President Lovett was adopted in 1958, with the East Hall residence becoming Baker College, South Hall residence becoming Will Rice College, West Hall becoming Hanszen College, and the temporary Wiess Hall becoming Wiess College.

    In 1959, the Rice Institute Computer went online. 1960 saw Rice Institute formally renamed William Marsh Rice University. Rice acted as a temporary intermediary in the transfer of land between Humble Oil and Refining Company and NASA, for the creation of NASA’s Manned Spacecraft Center (now called Johnson Space Center) in 1962. President John F. Kennedy then made a speech at Rice Stadium reiterating that the United States intended to reach the moon before the end of the decade of the 1960s, and “to become the world’s leading space-faring nation”. The relationship of NASA with Rice University and the city of Houston has remained strong to the present day.

    The original charter of Rice Institute dictated that the university admit and educate, tuition-free, “the white inhabitants of Houston, and the state of Texas”. In 1963, the governing board of Rice University filed a lawsuit to allow the university to modify its charter to admit students of all races and to charge tuition. Ph.D. student Raymond Johnson became the first black Rice student when he was admitted that year. In 1964, Rice officially amended the university charter to desegregate its graduate and undergraduate divisions. The Trustees of Rice University prevailed in a lawsuit to void the racial language in the trust in 1966. Rice began charging tuition for the first time in 1965. In the same year, Rice launched a $33 million ($268 million) development campaign. $43 million ($283 million) was raised by its conclusion in 1970. In 1974, two new schools were founded at Rice, the Jesse H. Jones Graduate School of Management and the Shepherd School of Music. The Brown Foundation Challenge, a fund-raising program designed to encourage annual gifts, was launched in 1976 and ended in 1996 having raised $185 million. The Rice School of Social Sciences was founded in 1979.

    On-campus housing was exclusively for men for the first forty years, until 1957. Jones College was the first women’s residence on the Rice campus, followed by Brown College. According to legend, the women’s colleges were purposefully situated at the opposite end of campus from the existing men’s colleges as a way of preserving campus propriety, which was greatly valued by Edgar Odell Lovett, who did not even allow benches to be installed on campus, fearing that they “might lead to co-fraternization of the sexes”. The path linking the north colleges to the center of campus was given the tongue-in-cheek name of “Virgin’s Walk”. Individual colleges became coeducational between 1973 and 1987, with the single-sex floors of colleges that had them becoming co-ed by 2006. By then, several new residential colleges had been built on campus to handle the university’s growth, including Lovett College, Sid Richardson College, and Martel College.

    Late twentieth and early twenty-first century

    The Economic Summit of Industrialized Nations was held at Rice in 1990. Three years later, in 1993, the James A. Baker III Institute for Public Policy was created. In 1997, the Edythe Bates Old Grand Organ and Recital Hall and the Center for Nanoscale Science and Technology, renamed in 2005 for the late Nobel Prize winner and Rice professor Richard E. Smalley, were dedicated at Rice. In 1999, the Center for Biological and Environmental Nanotechnology was created. The Rice Owls baseball team was ranked #1 in the nation for the first time in that year (1999), holding the top spot for eight weeks.

    In 2003, the Owls won their first national championship in baseball, which was the first for the university in any team sport, beating Southwest Missouri State in the opening game and then the University of Texas and Stanford University twice each en route to the title. In 2008, President David Leebron issued a ten-point plan titled “Vision for the Second Century” outlining plans to increase research funding, strengthen existing programs, and increase collaboration. The plan has brought about another wave of campus constructions, including the erection the newly renamed BioScience Research Collaborative building (intended to foster collaboration with the adjacent Texas Medical Center), a new recreational center and the renovated Autry Court basketball stadium, and the addition of two new residential colleges, Duncan College and McMurtry College.

    Beginning in late 2008, the university considered a merger with Baylor College of Medicine, though the merger was ultimately rejected in 2010. Rice undergraduates are currently guaranteed admission to Baylor College of Medicine upon graduation as part of the Rice/Baylor Medical Scholars program. According to History Professor John Boles’ recent book University Builder: Edgar Odell Lovett and the Founding of the Rice Institute, the first president’s original vision for the university included hopes for future medical and law schools.

    In 2018, the university added an online MBA program, MBA@Rice.

    In June 2019, the university’s president announced plans for a task force on Rice’s “past in relation to slave history and racial injustice”, stating that “Rice has some historical connections to that terrible part of American history and the segregation and racial disparities that resulted directly from it”.


    Rice’s campus is a heavily wooded 285-acre (115-hectare) tract of land in the museum district of Houston, located close to the city of West University Place.

    Five streets demarcate the campus: Greenbriar Street, Rice Boulevard, Sunset Boulevard, Main Street, and University Boulevard. For most of its history, all of Rice’s buildings have been contained within this “outer loop”. In recent years, new facilities have been built close to campus, but the bulk of administrative, academic, and residential buildings are still located within the original pentagonal plot of land. The new Collaborative Research Center, all graduate student housing, the Greenbriar building, and the Wiess President’s House are located off-campus.

    Rice prides itself on the amount of green space available on campus; there are only about 50 buildings spread between the main entrance at its easternmost corner, and the parking lots and Rice Stadium at the West end. The Lynn R. Lowrey Arboretum, consisting of more than 4000 trees and shrubs (giving birth to the legend that Rice has a tree for every student), is spread throughout the campus.
    The university’s first president, Edgar Odell Lovett, intended for the campus to have a uniform architecture style to improve its aesthetic appeal. To that end, nearly every building on campus is noticeably Byzantine in style, with sand and pink-colored bricks, large archways and columns being a common theme among many campus buildings. Noteworthy exceptions include the glass-walled Brochstein Pavilion, Lovett College with its Brutalist-style concrete gratings, Moody Center for the Arts with its contemporary design, and the eclectic-Mediterranean Duncan Hall. In September 2011, Travel+Leisure listed Rice’s campus as one of the most beautiful in the United States.

    The university and Houston Independent School District jointly established The Rice School-a kindergarten through 8th grade public magnet school in Houston. The school opened in August 1994. Through Cy-Fair ISD Rice University offers a credit course based summer school for grades 8 through 12. They also have skills based classes during the summer in the Rice Summer School.

    Innovation District

    In early 2019 Rice announced the site where the abandoned Sears building in Midtown Houston stood along with its surrounding area would be transformed into the “The Ion” the hub of the 16-acre South Main Innovation District. President of Rice David Leebron stated “We chose the name Ion because it’s from the Greek ienai, which means ‘go’. We see it as embodying the ever-forward motion of discovery, the spark at the center of a truly original idea.”

    Students of Rice and other Houston-area colleges and universities making up the Student Coalition for a Just and Equitable Innovation Corridor are advocating for a Community Benefits Agreement (CBA)-a contractual agreement between a developer and a community coalition. Residents of neighboring Third Ward and other members of the Houston Coalition for Equitable Development Without Displacement (HCEDD) have faced consistent opposition from the City of Houston and Rice Management Company to a CBA as traditionally defined in favor of an agreement between the latter two entities without a community coalition signatory.


    Rice University is chartered as a non-profit organization and is governed by a privately appointed board of trustees. The board consists of a maximum of 25 voting members who serve four-year terms. The trustees serve without compensation and a simple majority of trustees must reside in Texas including at least four within the greater Houston area. The board of trustees delegates its power by appointing a president to serve as the chief executive of the university. David W. Leebron was appointed president in 2004 and succeeded Malcolm Gillis who served since 1993. The provost six vice presidents and other university officials report to the president. The president is advised by a University Council composed of the provost, eight members of the Faculty Council, two staff members, one graduate student, and two undergraduate students. The president presides over a Faculty Council which has the authority to alter curricular requirements, establish new degree programs, and approve candidates for degrees.

    The university’s academics are organized into several schools. Schools that have undergraduate and graduate programs include:

    The Rice University School of Architecture
    The George R. Brown School of Engineering
    The School of Humanities
    The Shepherd School of Music
    The Wiess School of Natural Sciences
    The Rice University School of Social Sciences

    Two schools have only graduate programs:

    The Jesse H. Jones Graduate School of Management
    The Susanne M. Glasscock School of Continuing Studies

    Rice’s undergraduate students benefit from a centralized admissions process which admits new students to the university as a whole, rather than a specific school (the schools of Music and Architecture are decentralized). Students are encouraged to select the major path that best suits their desires; a student can later decide that they would rather pursue study in another field or continue their current coursework and add a second or third major. These transitions are designed to be simple at Rice with students not required to decide on a specific major until their sophomore year of study.

    Rice’s academics are organized into six schools which offer courses of study at the graduate and undergraduate level, with two more being primarily focused on graduate education, while offering select opportunities for undergraduate students. Rice offers 360 degrees in over 60 departments. There are 40 undergraduate degree programs, 51 masters programs, and 29 doctoral programs.

    Faculty members of each of the departments elect chairs to represent the department to each School’s dean and the deans report to the Provost who serves as the chief officer for academic affairs.

    Rice Management Company

    The Rice Management Company manages the $6.5 billion Rice University endowment (June 2019) and $957 million debt. The endowment provides 40% of Rice’s operating revenues. Allison Thacker is the President and Chief Investment Officer of the Rice Management Company, having joined the university in 2011.


    Rice is a medium-sized highly residential research university. The majority of enrollments are in the full-time four-year undergraduate program emphasizing arts & sciences and professions. There is a high graduate coexistence with the comprehensive graduate program and a very high level of research activity. It is accredited by the Southern Association of Colleges and Schools Commission on Colleges as well as the professional accreditation agencies for engineering, management, and architecture.

    Each of Rice’s departments is organized into one of three distribution groups, and students whose major lies within the scope of one group must take at least 3 courses of at least 3 credit hours each of approved distribution classes in each of the other two groups, as well as completing one physical education course as part of the LPAP (Lifetime Physical Activity Program) requirement. All new students must take a Freshman Writing Intensive Seminar (FWIS) class, and for students who do not pass the university’s writing composition examination (administered during the summer before matriculation), FWIS 100, a writing class, becomes an additional requirement.

    The majority of Rice’s undergraduate degree programs grant B.S. or B.A. degrees. Rice has recently begun to offer minors in areas such as business, energy and water sustainability, and global health.

    Student body

    As of fall 2014, men make up 52% of the undergraduate body and 64% of the professional and post-graduate student body. The student body consists of students from all 50 states, including the District of Columbia, two U.S. Territories, and 83 foreign countries. Forty percent of degree-seeking students are from Texas.

    Research centers and resources

    Rice is noted for its applied science programs in the fields of nanotechnology, artificial heart research, structural chemical analysis, signal processing and space science.

    Rice Alliance for Technology and Entrepreneurship – supports entrepreneurs and early-stage technology ventures in Houston and Texas through education, collaboration, and research, ranked No. 1 among university business incubators.
    Baker Institute for Public Policy – a leading nonpartisan public policy think-tank
    BioScience Research Collaborative (BRC) – interdisciplinary, cross-campus, and inter-institutional resource between Rice University and Texas Medical Center
    Boniuk Institute – dedicated to religious tolerance and advancing religious literacy, respect and mutual understanding
    Center for African and African American Studies – fosters conversations on topics such as critical approaches to race and racism, the nature of diasporic histories and identities, and the complexity of Africa’s past, present and future
    Chao Center for Asian Studies – research hub for faculty, students and post-doctoral scholars working in Asian studies
    Center for the Study of Women, Gender, and Sexuality (CSWGS) – interdisciplinary academic programs and research opportunities, including the journal Feminist Economics
    Data to Knowledge Lab (D2K) – campus hub for experiential learning in data science
    Digital Signal Processing (DSP) – center for education and research in the field of digital signal processing
    Ethernest Hackerspace – student-run hackerspace for undergraduate engineering students sponsored by the ECE department and the IEEE student chapter
    Humanities Research Center (HRC) – identifies, encourages, and funds innovative research projects by faculty, visiting scholars, graduate, and undergraduate students in the School of Humanities and beyond
    Institute of Biosciences and Bioengineering (IBB) – facilitates the translation of interdisciplinary research and education in biosciences and bioengineering
    Ken Kennedy Institute for Information Technology – advances applied interdisciplinary research in the areas of computation and information technology
    Kinder Institute for Urban Research – conducts the Houston Area Survey, “the nation’s longest running study of any metropolitan region’s economy, population, life experiences, beliefs and attitudes”
    Laboratory for Nanophotonics (LANP) – a resource for education and research breakthroughs and advances in the broad, multidisciplinary field of nanophotonics
    Moody Center for the Arts – experimental arts space featuring studio classrooms, maker space, audiovisual editing booths, and a gallery and office space for visiting national and international artists
    OpenStax CNX (formerly Connexions) and OpenStax – an open source platform and open access publisher, respectively, of open educational resources
    Oshman Engineering Design Kitchen (OEDK) – space for undergraduate students to design, prototype and deploy solutions to real-world engineering challenges
    Rice Cinema – an independent theater run by the Visual and Dramatic Arts department at Rice which screens documentaries, foreign films, and experimental cinema and hosts film festivals and lectures since 1970
    Rice Center for Engineering Leadership (RCEL) – inspires, educates, and develops ethical leaders in technology who will excel in research, industry, non-engineering career paths, or entrepreneurship
    Religion and Public Life Program (RPLP) – a research, training and outreach program working to advance understandings of the role of religion in public life
    Rice Design Alliance (RDA) – outreach and public programs of the Rice School of Architecture
    Rice Center for Quantum Materials (RCQM) – organization dedicated to research and higher education in areas relating to quantum phenomena
    Rice Neuroengineering Initiative (NEI) – fosters research collaborations in neural engineering topics
    Rice Space Institute (RSI) – fosters programs in all areas of space research
    Smalley-Curl Institute for Nanoscale Science and Technology (SCI) – the nation’s first nanotechnology center
    Welch Institute for Advanced Materials – collaborative research institute to support the foundational research for discoveries in materials science, similar to the model of Salk Institute and Broad Institute
    Woodson Research Center Special Collections & Archives – publisher of print and web-based materials highlighting the department’s primary source collections such as the Houston African American, Asian American, and Jewish History Archives, University Archives, rare books, and hip hop/rap music-related materials from the Swishahouse record label and Houston Folk Music Archive, etc.

    Residential colleges

    In 1957, Rice University implemented a residential college system, which was proposed by the university’s first president, Edgar Odell Lovett. The system was inspired by existing systems in place at University of Oxford (UK) and University of Cambridge (UK) and at several other universities in the United States, most notably Yale University. The existing residences known as East, South, West, and Wiess Halls became Baker, Will Rice, Hanszen, and Wiess Colleges, respectively.

    Student-run media

    Rice has a weekly student newspaper (The Rice Thresher), a yearbook (The Campanile), college radio station (KTRU Rice Radio), and now defunct, campus-wide student television station (RTV5). They are based out of the RMC student center. In addition, Rice hosts several student magazines dedicated to a range of different topics; in fact, the spring semester of 2008 saw the birth of two such magazines, a literary sex journal called Open and an undergraduate science research magazine entitled Catalyst.

    The Rice Thresher is published every Wednesday and is ranked by Princeton Review as one of the top campus newspapers nationally for student readership. It is distributed around campus, and at a few other local businesses and has a website. The Thresher has a small, dedicated staff and is known for its coverage of campus news, open submission opinion page, and the satirical Backpage, which has often been the center of controversy. The newspaper has won several awards from the College Media Association, Associated Collegiate Press and Texas Intercollegiate Press Association.

    The Rice Campanile was first published in 1916 celebrating Rice’s first graduating class. It has published continuously since then, publishing two volumes in 1944 since the university had two graduating classes due to World War II. The website was created sometime in the early to mid 2000’s. The 2015 won the first place Pinnacle for best yearbook from College Media Association.

    KTRU Rice Radio is the student-run radio station. Though most DJs are Rice students, anyone is allowed to apply. It is known for playing genres and artists of music and sound unavailable on other radio stations in Houston, and often, the US. The station takes requests over the phone or online. In 2000 and 2006, KTRU won Houston Press’ Best Radio Station in Houston. In 2003, Rice alum and active KTRU DJ DL’s hip-hip show won Houston PressBest Hip-hop Radio Show. On August 17, 2010, it was announced that Rice University had been in negotiations to sell the station’s broadcast tower, FM frequency and license to the University of Houston System to become a full-time classical music and fine arts programming station. The new station, KUHA, would be operated as a not-for-profit outlet with listener supporters. The FCC approved the sale and granted the transfer of license to the University of Houston System on April 15, 2011, however, KUHA proved to be an even larger failure and so after four and a half years of operation, The University of Houston System announced that KUHA’s broadcast tower, FM frequency and license were once again up for sale in August 2015. KTRU continued to operate much as it did previously, streaming live on the Internet, via apps, and on HD2 radio using the 90.1 signal. Under student leadership, KTRU explored the possibility of returning to FM radio for a number of years. In spring 2015, KTRU was granted permission by the FCC to begin development of a new broadcast signal via LPFM radio. On October 1, 2015, KTRU made its official return to FM radio on the 96.1 signal. While broadcasting on HD2 radio has been discontinued, KTRU continues to broadcast via internet in addition to its LPFM signal.

    RTV5 is a student-run television network available as channel 5 on campus. RTV5 was created initially as Rice Broadcast Television in 1997; RBT began to broadcast the following year in 1998, and aired its first live show across campus in 1999. It experienced much growth and exposure over the years with successful programs like Drinking with Phil, The Meg & Maggie Show, which was a variety and call-in show, a weekly news show, and extensive live coverage in December 2000 of the shut down of KTRU by the administration. In spring 2001, the Rice undergraduate community voted in the general elections to support RBT as a blanket tax organization, effectively providing a yearly income of $10,000 to purchase new equipment and provide the campus with a variety of new programming. In the spring of 2005, RBT members decided the station needed a new image and a new name: Rice Television 5. One of RTV5’s most popular shows was the 24-hour show, where a camera and couch placed in the RMC stayed on air for 24 hours. One such show is held in fall and another in spring, usually during a weekend allocated for visits by prospective students. RTV5 has a video on demand site at rtv5.rice.edu. The station went off the air in 2014 and changed its name to Rice Video Productions. In 2015 the group’s funding was threatened, but ultimately maintained. In 2016 the small student staff requested to no longer be a blanket-tax organization. In the fall of 2017, the club did not register as a club.

    The Rice Review, also known as R2, is a yearly student-run literary journal at Rice University that publishes prose, poetry, and creative nonfiction written by undergraduate students, as well as interviews. The journal was founded in 2004 by creative writing professor and author Justin Cronin.

    The Rice Standard was an independent, student-run variety magazine modeled after such publications as The New Yorker and Harper’s. Prior to fall 2009, it was regularly published three times a semester with a wide array of content, running from analyses of current events and philosophical pieces to personal essays, short fiction and poetry. In August 2009, The Standard transitioned to a completely online format with the launch of their redesigned website, http://www.ricestandard.org. The first website of its kind on Rice’s campus, The Standard featured blog-style content written by and for Rice students. The Rice Standard had around 20 regular contributors, and the site features new content every day (including holidays). In 2017 no one registered The Rice Standard as a club within the university.

    Open, a magazine dedicated to “literary sex content,” predictably caused a stir on campus with its initial publication in spring 2008. A mixture of essays, editorials, stories and artistic photography brought Open attention both on campus and in the Houston Chronicle. The third and last annual edition of Open was released in spring of 2010.


    Rice plays in NCAA Division I athletics and is part of Conference USA. Rice was a member of the Western Athletic Conference before joining Conference USA in 2005. Rice is the second-smallest school, measured by undergraduate enrollment, competing in NCAA Division I FBS football, only ahead of Tulsa.

    The Rice baseball team won the 2003 College World Series, defeating Stanford, giving Rice its only national championship in a team sport. The victory made Rice University the smallest school in 51 years to win a national championship at the highest collegiate level of the sport. The Rice baseball team has played on campus at Reckling Park since the 2000 season. As of 2010, the baseball team has won 14 consecutive conference championships in three different conferences: the final championship of the defunct Southwest Conference, all nine championships while a member of the Western Athletic Conference, and five more championships in its first five years as a member of Conference USA. Additionally, Rice’s baseball team has finished third in both the 2006 and 2007 College World Series tournaments. Rice now has made six trips to Omaha for the CWS. In 2004, Rice became the first school ever to have three players selected in the first eight picks of the MLB draft when Philip Humber, Jeff Niemann, and Wade Townsend were selected third, fourth, and eighth, respectively. In 2007, Joe Savery was selected as the 19th overall pick.

    Rice has been very successful in women’s sports in recent years. In 2004–05, Rice sent its women’s volleyball, soccer, and basketball teams to their respective NCAA tournaments. The women’s swim team has consistently brought at least one member of their team to the NCAA championships since 2013. In 2005–06, the women’s soccer, basketball, and tennis teams advanced, with five individuals competing in track and field. In 2006–07, the Rice women’s basketball team made the NCAA tournament, while again five Rice track and field athletes received individual NCAA berths. In 2008, the women’s volleyball team again made the NCAA tournament. In 2011 the Women’s Swim team won their first conference championship in the history of the university. This was an impressive feat considering they won without having a diving team. The team repeated their C-USA success in 2013 and 2014. In 2017, the women’s basketball team, led by second-year head coach Tina Langley, won the Women’s Basketball Invitational, defeating UNC-Greensboro 74–62 in the championship game at Tudor Fieldhouse. Though not a varsity sport, Rice’s ultimate frisbee women’s team, named Torque, won consecutive Division III national championships in 2014 and 2015.

    In 2006, the football team qualified for its first bowl game since 1961, ending the second-longest bowl drought in the country at the time. On December 22, 2006, Rice played in the New Orleans Bowl in New Orleans, Louisiana against the Sun Belt Conference champion, Troy. The Owls lost 41–17. The bowl appearance came after Rice had a 14-game losing streak from 2004–05 and went 1–10 in 2005. The streak followed an internally authorized 2003 McKinsey report that stated football alone was responsible for a $4 million deficit in 2002. Tensions remained high between the athletic department and faculty, as a few professors who chose to voice their opinion were in favor of abandoning the football program. The program success in 2006, the Rice Renaissance, proved to be a revival of the Owl football program, quelling those tensions. David Bailiff took over the program in 2007 and has remained head coach. Jarett Dillard set an NCAA record in 2006 by catching a touchdown pass in 13 consecutive games and took a 15-game overall streak into the 2007 season.

    In 2008, the football team posted a 9-3 regular season, capping off the year with a 38–14 victory over Western Michigan University in the Texas Bowl. The win over Western Michigan marked the Owls’ first bowl win in 45 years.

    Rice Stadium also serves as the performance venue for the university’s Marching Owl Band, or “MOB.” Despite its name, the MOB is a scatter band that focuses on performing humorous skits and routines rather than traditional formation marching.

    Rice Owls men’s basketball won 10 conference titles in the former Southwest Conference (1918, 1935*, 1940, 1942*, 1943*, 1944*, 1945, 1949*, 1954*, 1970; * denotes shared title). Most recently, guard Morris Almond was drafted in the first round of the 2007 NBA Draft by the Utah Jazz. Rice named former Cal Bears head coach Ben Braun as head basketball coach to succeed Willis Wilson, fired after Rice finished the 2007–2008 season with a winless (0-16) conference record and overall record of 3-27.

  • richardmitnick 5:25 pm on February 20, 2023 Permalink | Reply
    Tags: "How will AI change mathematics? Rise of chatbots highlights discussion", , Machine learning tools already help mathematicians to formulate new theories and solve tough problems. But they’re set to shake up the field even more., Mathematics,   

    From “Nature” : “How will AI change mathematics? Rise of chatbots highlights discussion” 

    From “Nature”

    Davide Castelvecchi

    Machine learning tools already help mathematicians to formulate new theories and solve tough problems. But they’re set to shake up the field even more.

    AI tools have allowed researchers to solve complex mathematical problems.Credit: Fadel Senna/AFP/Getty.

    As interest in chatbots spreads like wildfire, mathematicians are beginning to explore how artificial intelligence (AI) could help them to do their work. Whether it’s assisting with verifying human-written work or suggesting new ways to solve difficult problems, automation is beginning to change the field in ways that go beyond mere calculation, researchers say.

    “We’re looking at a very specific question: will machines change Math?” says Andrew Granville, a number theorist at the University of Montreal (CA). A workshop at the University of California-Los Angeles this week explored this question, aiming to build bridges between mathematicians and computer scientists. “Most mathematicians are completely unaware of these opportunities,” says one of the event’s organizers, Marijn Heule, a computer scientist at Carnegie Mellon University in Pittsburgh.

    Akshay Venkatesh, a 2018 winner of the prestigious Fields Medal who is at the Institute for Advanced Study in Princeton kick-started a conversation on how computers will change Maths at a symposium in his honour in October. Two other recipients of the medal, Timothy Gowers at the Collège de France (FR) and Terence Tao at UCLA, have also taken leading roles in the debate.

    “The fact that we have people like Fields medalists and other very famous big-shot mathematicians interested in the area now is an indication that it’s ‘hot’ in a way that it didn’t used to be,” says Kevin Buzzard, a mathematician at Imperial College London (UK).

    AI approaches

    Part of the discussion concerns what kind of automation tools will be most useful. AI comes in two major flavors. In ‘symbolic’ AI, programmers embed rules of logic or calculation into their code. “It’s what people would call ‘good old-fashioned AI’,” says Leonardo de Moura, a computer scientist at Microsoft Research in Redmond, Washington.

    The other approach, which has become extremely successful in the past decade or so, is based on artificial neural networks. In this type of AI, the computer starts more or less from a clean slate and learns patterns by digesting large amounts of data. This is called machine-learning, and it is the basis of ‘large language models’ (including chatbots such as ChatGPT), as well as the systems that can beat human players at complex games or predict how proteins fold. Whereas symbolic AI is inherently rigorous, neural networks can only make statistical guesses, and their operations are often mysterious.

    De Moura helped symbolic AI to score some early mathematical successes by creating a system called Lean. This interactive software tool forces researchers to write out each logical step of a problem, down to the most basic details, and ensures that the maths is correct. Two years ago, a team of mathematicians succeeded at translating an important but impenetrable proof — one so complicated that even its author was unsure of it — into Lean, thereby confirming that it was correct.

    The researchers say the process helped them to understand the proof, and even to find ways to simplify it. “I think this is even more exciting than checking the correctness,” de Moura says. “Even in our wildest dreams, we didn’t imagine that.”

    As well as making solitary work easier, this sort of ‘proof assistant’ could change how mathematicians work together by eliminating what de Moura calls a “trust bottleneck”. “When we are collaborating, I may not trust what you are doing. But a proof assistant shows your collaborators that they can trust your part of the work.”

    Sophisticated autocomplete

    At the other extreme are chatbot-esque, neural-network-based large language models. At Google, former physicist Ethan Dyer and his team have developed a chatbot called Minerva, which specializes in solving maths problems. At heart, Minerva is a very sophisticated version of the autocomplete function on messaging apps: by training on maths papers in the arXiv repository, it has learnt to write down step-by-step solutions to problems in the same way that some apps can predict words and phrases. Unlike Lean, which communicates using something similar to computer code, Minerva takes questions and writes answers in conversational English. “It is an achievement to solve some of these problems automatically,” says de Moura.

    Minerva shows both the power and the possible limitations of this approach. For example, it can accurately factor integer numbers into primes — numbers that can’t be divided evenly into smaller ones. But it starts making mistakes once the numbers exceed a certain size, showing that it has not ‘understood’ the general procedure.

    Still, Minerva’s neural network seems to be able to acquire some general techniques, as opposed to just statistical patterns, and the Google team is trying to understand how it does that. “Ultimately, we’d like a model that you can brainstorm with,” Dyer says. He says it could also be useful for non-mathematicians who need to extract information from the specialized literature. Further extensions will expand Minerva’s skills by studying textbooks and interfacing with dedicated maths software.

    Dyer says the motivation behind the Minerva project was to see how far the machine-learning approach could be pushed; a powerful automated tool to help mathematicians might end up combining symbolic AI techniques with neural networks.

    Maths v. machines

    In the longer term, will programs remain part of the supporting cast, or will they be able to conduct mathematical research independently? AI might get better at producing correct mathematical statements and proofs, but some researchers worry that most of those would be uninteresting or impossible to understand. At the October symposium, Gowers said that there might be ways of teaching a computer some objective criteria for mathematical relevance, such as whether a small statement can embody many special cases or even form a bridge between different subfields of maths. “In order to get good at proving theorems, computers will have to judge what is interesting and worth proving,” he said. If they can do that, the future of humans in the field looks uncertain.

    Computer scientist Erika Abraham at RWTH Aachen University (DE) is more sanguine about the future of mathematicians. “An AI system is only as smart as we program it to be,” she says. “The intelligence is not in the computer; the intelligence is in the programmer or trainer.”

    Melanie Mitchell, a computer scientist and cognitive scientist at the Santa Fe Institute says that mathematicians’ jobs will be safe until a major shortcoming of AI is fixed — its inability to extract abstract concepts from concrete information. “While AI systems might be able to prove theorems, it’s much harder to come up with interesting mathematical abstractions that give rise to the theorems in the first place.”

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    ”Nature” is a weekly international journal publishing the finest peer-reviewed research in all fields of science and technology on the basis of its originality, importance, interdisciplinary interest, timeliness, accessibility, elegance and surprising conclusions. Nature also provides rapid, authoritative, insightful and arresting news and interpretation of topical and coming trends affecting science, scientists and the wider public.

  • richardmitnick 10:30 am on February 7, 2023 Permalink | Reply
    Tags: "Topology": the properties of a geometric shape that don’t change, "Unlocking the Secrets of a 4D Cosmos", , , , Knot theory: an intersection point where ideas from theoretical physics-like string theory-intersect with some of our more sophisticated mathematical tools., Mathematics, , The Dornsife College of Letters Arts and Sciences, The project could shed important new light on the theories for Quantum Gravity or a theory that combines General Relativity and Quantum Mechanics.,   

    From The Dornsife College of Letters Arts and Sciences At The University of Southern California: “Unlocking the Secrets of a 4D Cosmos” 

    From The Dornsife College of Letters Arts and Sciences


    USC bloc

    The University of Southern California

    Darrin S. Joy

    A USC Dornsife mathematician is leading a project that could turn science fiction dreams into reality, supported by 11 esteemed institutions and a landmark $8 million grant.

    A new collaboration aims to push the boundaries of knowledge about the fundamental character of four-dimensional space-time. (Composite: Rhiannon Montelius. Image Sources: Unsplash and PxHere.)

    The four-year project, funded by an $8 million Simons Foundation grant, brings together researchers from 11 prestigious institutions to tackle some of the most pressing questions in mathematics and theoretical physics.

    A key distinction of this type of funding: It aims to extend the boundaries of knowledge by assembling scientists, researchers and scholars who don’t typically interact.

    This story involves two kinds of math: simple and very complex.

    The simple math adds up to an exceptional amount of money — $8 million, in fact — to study the other, more complex math over the next four years. That’s $2 million per year, and a few more digits than are typically found on grants awarded for math research.

    Most grants funding science go to experimental equipment, postdoctoral scholars, graduate students and other expensive resources.

    “In math, all we typically need is a paper and pencil, so grants tend to be much less,” says Aaron Lauda, professor of mathematics and physics and astronomy at the USC Dornsife College of Letters, Arts and Sciences. Lauda led the charge to obtain the prestigious grant, which establishes the Simons Collaboration on New Structures in Low-Dimensional Topology.

    The grant is unique in that it is sufficiently large to fund a sizable team working in a similar direction and will support extended visits by team members and other experts that will enhance the research activities and community at USC, says Lauda, who directs the collaboration.

    The funding comes from the Simons Foundation. Lauda as director, along with grant co-principal investigator Cris Negron, assistant professor of mathematics, brings nearly 14% of the total — about $1.1 million — to USC. It’s one of the largest awards ever obtained for USC Dornsife’s math department.

    The remaining $7.9 million is divided among 10 other institutions: Caltech, the Massachusetts Institute of Technology, UCLA, the University of North Carolina at Chapel Hill, and Columbia, Stanford, Princeton and Harvard universities, as well as the Centre for Quantum Mathematics at Southern Denmark University and the University of Zurich.

    Simons collaboration aims to solve knotty math problems

    The complex math that Lauda and the other researchers are exploring centers on topology. In vastly oversimplified language, topology refers to the properties of a geometric shape that don’t change no matter how much you futz with it.

    Lauda points to knots as an example.

    A length of rope twisted and tangled, and then joined at the ends to make a single, contiguous knot, has its own topology. No matter how the knot is positioned, it’s still the same knot.

    “If I wiggle it around, as long as I don’t cut it or break it, it’s fundamentally the same,” Lauda says.

    If two people arbitrarily twist and tangle different but identical ropes of the same size and seal the ends together, even the most discerning eye might have difficulty determining if the two knots are the same or different.

    As Lauda explains it, we can try to wiggle one knot to look the same as the other, but if we don’t succeed, we can’t necessarily conclude that the two knots are different. Perhaps we didn’t try hard enough. This is where the mathematics of knot invariants can help.

    A knot invariant is a way of assigning a number or polynomial to a knot that stays the same no matter how the knot is wiggled or deformed. If our invariant assigns different numbers, or polynomials, to two different knots, then we can be certain that the two knots are different.

    Determining if different knots are topologically the same can be challenging. A) Two “knots” that are topologically the same. B) A third knot that is topologically the same as the first two. C) Two knots that differ topologically; the upper can be unknotted but the lower (the “trefoil knot”) cannot. (Images: Aaron Lauda.)

    Mathematicians push for advances on par with Einstein’s

    As fun as knots can be, Lauda and his project collaborators have set their sights on more grandiose issues of topology, aiming to build on existing theory and tools to answer fundamental questions about the universe.

    “Knot theory is interesting because it’s an intersection point where ideas from theoretical physics, like string theory, intersect with some of our more sophisticated mathematical tools,” Lauda says. “But what the collaboration is really trying to do goes way beyond knots, to understand the ways in which even our reality can have these types of topological properties.”

    Lauda refers to knowledge on par with that revealed by Albert Einstein and his General Theory of Relativity, namely that the three dimensions of space are intimately linked with a fourth dimension, that of time. And mass — planets, stars and even knots — warp space-time, something we experience as gravity.

    In revealing the gravitational curvature of space-time, Einstein revolutionized perceptions of the universe, and this in turn made technologies such as GPS navigation possible.

    Lauda says that, when it comes to topology, accounting for that fourth dimension has led to important advances in Math and Physics, and he and his collaborators aim to push the boundaries of knowledge.

    “Mathematically, it’s important to understand and classify four-dimensional topological objects in the same way we do knots,” he says. “That’s the sort of big frontier we’re trying to accomplish with this collaboration.”

    Math collaborators work toward advances in four dimensions

    Mathematicians trying to describe spacetime do so by specifying four dimensions: three spatial dimensions (up and down, back and forth, right and left) and one time dimension.

    For mathematicians and physicists, working on that fourth dimension proves most challenging, a fact Lauda finds intriguing. “It is a curious fact of nature that dimension four is unique in that our understanding is vastly limited in precisely this dimension that just so happens to coincide with the four dimensions that compose our universe.”

    The Simons Collaboration aims to tackle this confounding dimension head on, developing the next generation of tools for studying four-dimensional topology, unlocking new techniques arising from Theoretical Physics and resolving long-standing questions that have plagued mathematicians for decades, Lauda says.

    For example, the project could shed important new light on the theories for quantum gravity, or a theory that combines general relativity and quantum physics.

    “General relativity works fantastically when you’re talking about planets, stars and the like,” Lauda explains. “Quantum mechanics works when you’re talking about very small things like subatomic particles. But once you start combining those small things with the big things, like one might encounter near a black hole, or near the big bang, the theories start breaking down.”

    He says the project’s efforts could set the stage for finally closing the divide, giving deeper insight into how the universe works at all levels.

    What’s more, any breakthroughs in understanding could lead to scientific and technological advances in the same way Einstein’s discoveries led to GPS, he says.

    “What we expect to learn from this project could be the foundation for future technologies and advances that seem like pure science fiction now.”

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    The USC Dornsife College of Letters, Arts and Sciences is the academic core of the University of Southern California. Our diverse community works across the natural sciences, social sciences and humanities, exploring fundamental questions about who we are, how the world works, and what we can do to improve and enrich society.

    Our faculty push the boundaries of knowledge and equip students with lifelong learning skills that enable them to overcome challenges of extraordinary social and technical complexity. Beyond the classrooms and laboratories, our scholars work hand-in-hand with leaders in the public and private sectors — bringing new ways of thinking to projects that call for their expertise.

    Some of the most talented students in the world are attracted to USC Dornsife for its innovative courses and customizable degree programs. And because they are not content to wait until graduation to make an impact, every undergraduate takes part in hands-on experiential learning that converts thought to action.

    Our USC Dornsife community values the insight we gain from our position in one of the most dynamic and diverse cities on Earth. Los Angeles is where cultures integrate, ideas are born, and trends are set. It is a place where global issues are expressed locally, allowing us to innovate the solutions here first.

    The future belongs to creative thinkers. At the USC Dornsife College of Letters, Arts and Sciences, we enable their creativity to flourish.

    USC campus

    The The University of Southern California is a private research university in Los Angeles, California. Founded in 1880 by Robert M. Widney, it is the oldest private research university in California.

    The university is composed of one liberal arts school, the Dornsife College of Letters, Arts and Sciences and twenty-two undergraduate, graduate and professional schools, enrolling an average of 19,500 undergraduate and 26,500 post-graduate students from all fifty U.S. states and more than 115 countries. USC is ranked among the top universities in the United States and admission to its programs is highly selective.

    USC is a member of The Association of American Universities, joining in 1969. The University of Southern California houses professional schools offering a number of varying disciplines among which include communication, law, dentistry, medicine, business, engineering, journalism, public policy, music, architecture, and cinematic arts. USC’s academic departments fall either under the general liberal arts and sciences of the College of Letters, Arts, and Sciences for undergraduates, the Graduate School for graduates, or the university’s 17 professional schools.

    USC was one of the earliest nodes on ARPANET and is the birthplace of the Domain Name System. Other technologies invented at USC include DNA computing, dynamic programming, image compression, VoIP, and antivirus software.

    USC’s notable alumni include 11 Rhodes scholars and 12 Marshall scholars. As of January 2021, 10 Nobel laureates, six MacArthur Fellows, and one Turing Award winner have been affiliated with the university. USC has conferred degrees upon 29 alumni who became billionaires, and has graduated more alumni who have gone on to win Academy and Emmy Awards than any other institution in the world by a significant margin, in part due to the success of the School of Cinematic Arts.

    USC sponsors a variety of intercollegiate sports and competes in the National Collegiate Athletic Association (NCAA) as a member of the Pac-12 Conference. Members of USC’s sports teams, the Trojans, have won 107 NCAA team championships, ranking them third in the United States, and 412 NCAA individual championships, ranking them third in the United States and second among NCAA Division I schools. Trojan athletes have won 309 medals at the Olympic Games (144 golds, 93 silvers and 72 bronzes), more than any other university in the United States. In 1969, it joined the Association of American Universities. USC has had a total of 537 football players drafted to the National Football League, the second-highest number of drafted players in the country.

    The University of Southern California is the largest private employer in the Los Angeles area and generates an estimated $8 billion of economic impact on California.

    Faculty and Research

    The university is classified among “R1: Doctoral Universities – Very high research activity”. According to the National Science Foundation, USC spent $891 million on research and development in 2018, ranking it 23rd in the nation.

    USC employs approximately 4,706 full-time faculty, 1,816 part-time faculty, 16,614 staff members, and 4,817 student workers. 350 postdoctoral fellows are supported along with over 800 medical residents. Among the USC faculty, 17 are members of the National Academy of Sciences, 16 are members of the National Academy of Medicine, 37 are members of the National Academy of Engineering, 97 are members of the American Association for the Advancement of Science, and 34 are members of the American Academy of Arts and Sciences, 5 to the American Philosophical Society, and 14 to the National Academy of Public Administration . 29 USC faculty are listed as among the “Highly Cited” in the Institute for Scientific Information database. George Olah won the 1994 Nobel Prize in Chemistry and was the founding director of the Loker Hydrocarbon Research Institute. Leonard Adleman won the Turing Award in 2003. Arieh Warshel won the 2013 Nobel Prize in Chemistry.

    The university also supports the Pacific Council on International Policy through joint programming, leadership collaboration, and facilitated connections among students, faculty, and Pacific Council members.

    The university has two National Science Foundation–funded Engineering Research Centers: The Integrated Media Systems Center and the Center for Biomimetic Microelectronic Systems. The Department of Homeland Security selected USC as its first Homeland Security Center of Excellence. Since 1991, USC has been the headquarters of the NSF and USGS funded Southern California Earthquake Center (SCEC). The University of Southern California is a founding and charter member of CENIC, the Corporation for Education Network Initiatives in California, the nonprofit organization, which provides extremely high-performance Internet-based networking to California’s K-20 research and education community. USC researcher Jonathan Postel was an editor of communications-protocol for the fledgling internet, also known as ARPANET.

    In July 2016 USC became home to the world’s most powerful quantum computer, housed in a super-cooled, magnetically shielded facility at the USC Information Sciences Institute, the only other commercially available quantum computing system operated jointly by National Aeronautics Space Agency and Google.

    Notable USC faculty include or have included the following: Leonard Adleman, Richard Bellman, Aimee Bender, Barry Boehm, Warren Bennis, Todd Boyd, T.C. Boyle, Leo Buscaglia, Drew Casper, Manuel Castells, Erwin Chemerinsky, George V. Chilingar, Thomas Crow, António Damásio, Francis De Erdely, Percival Everett, Murray Gell-Mann, Seymour Ginsburg, G. Thomas Goodnight, Jane Goodall, Solomon Golomb, Midori Goto, Susan Estrich, Janet Fitch, Tomlinson Holman, Jascha Heifetz, Henry Jenkins, Thomas H. Jordan, Mark Kac, Pierre Koenig, Neil Leach, Leonard Maltin, Daniel L. McFadden, Viet Thanh Nguyen, George Olah, Scott Page, Tim Page (music critic), Simon Ramo, Claudia Rankine, Irving Reed, Michael Waterman, Frank Gehry, Arieh Warshel, Lloyd Welch, Jonathan Taplin, and Diane Winston.

    • Lemuel Radar 6:31 pm on March 2, 2023 Permalink | Reply

      I just wanted to say something and let you know how much I appreciated reading your website. The content are fantastic and have given me new ideas. I’ll be adding it to my list and look forward to reading your future posts. Thank you for sharing your knowledge and continue the fantastic work!


  • richardmitnick 1:40 pm on January 16, 2023 Permalink | Reply
    Tags: "A Teenager Solved a Stubborn Prime Number ‘Look-Alike’ Riddle", "Carmichael numbers"—strange entities that mimic the primes., , Daniel Larsen, For more than a year and a half Larsen couldn’t stop thinking about a certain math problem: "Carmichael numbers"., He gets focused on something and it’s just bang bang bang until he succeeds., It was only in the mid-1990s that mathematicians proved there are infinitely many of them., Larsen holds the record for youngest person to publish a crossword in "The New York Times"., Mathematics, Over a century ago in that quest for a fast powerful primality test mathematicians stumbled on a group of troublemakers—numbers that fool tests into thinking they’re prime even though they’re no, These pseudoprimes known as "Carmichael numbers" have been particularly difficult to grasp.,   

    From “WIRED”: “A Teenager Solved a Stubborn Prime Number ‘Look-Alike’ Riddle” Daniel Larsen 

    From “WIRED”

    Jordana Cepelewicz

    In his senior year of high school, Daniel Larsen proved a key theorem about “Carmichael numbers”—strange entities that mimic the primes.

    After he posted his proof, Daniel Larsen enrolled at the Massachusetts Institute of Technology as a math major. Photograph: Katherine Taylor/Quanta Magazine.

    When Daniel Larsen was in middle school, he started designing crossword puzzles. He had to layer the hobby on top of his other interests: chess, programming, piano, violin. He twice qualified for the Scripps National Spelling Bee near Washington, DC, after winning his regional competition. “He gets focused on something, and it’s just bang, bang, bang, until he succeeds,” said Larsen’s mother, Ayelet Lindenstrauss. His first crossword puzzles were rejected by major newspapers, but he kept at it and ultimately broke in. To date, he holds the record for youngest person to publish a crossword in The New York Times, at age 13. “He’s very persistent,” Lindenstrauss said.

    Still, Larsen’s most recent obsession felt different, “longer and more intense than most of his other projects,” she said. For more than a year and a half Larsen couldn’t stop thinking about a certain math problem.

    It had roots in a broader question, one that the mathematician Carl Friedrich Gauss considered to be among the most important in mathematics: how to distinguish a prime number (a number that is divisible only by 1 and itself) from a composite number. For hundreds of years, mathematicians have sought an efficient way to do so. The problem has also become relevant in the context of modern cryptography, as some of today’s most widely used cryptosystems involve doing arithmetic with enormous primes.

    Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers—numbers that fool tests into thinking they’re prime, even though they’re not. These pseudoprimes known as “Carmichael numbers” have been particularly difficult to grasp. It was only in the mid-1990s, for instance, that mathematicians proved there are infinitely many of them. Being able to say something more about how they’re distributed along the number line has posed an even greater challenge.

    Then along came Larsen with a new proof about just that, one inspired by recent epochal work in a different area of number theory. At the time, he was just 17.

    The Spark

    Growing up in Bloomington, Indiana, Larsen was always drawn to mathematics. His parents, both mathematicians, introduced him and his older sister to the subject when they were young. (His sister is now pursuing a doctorate in math.) When Larsen was 3 years old, Lindenstrauss recalls, he started asking her philosophical questions about the nature of infinity. “I thought, this kid has a mathematical mind,” said Lindenstrauss, a professor at Indiana University.

    Then a few years ago—around the time that he was immersed in his spelling and crossword projects—he came across a documentary about Yitang Zhang, an unknown mathematician who rose from obscurity in 2013 after proving a landmark result that put an upper bound on the gaps between consecutive prime numbers. Something clicked in Larsen. He couldn’t stop thinking about number theory, and about the related problem that Zhang and other mathematicians still hoped to solve: the twin primes conjecture, which states that there are infinitely many pairs of primes that differ by only 2.

    The High Schooler Who Solved a Prime Number Theorem. Daniel Larsen wouldn’t let go of an old question about Carmichael numbers. “It was just stubbornness on my part,” he said.

    After Zhang’s work, which showed that there are infinitely many pairs of primes that differ by less than 70 million, others jumped in to lower this bound even further. Within months, the mathematicians James Maynard and Terence Tao independently proved an even stronger statement about the gaps between primes. That gap has since shrunk to 246.

    Larsen wanted to understand some of the mathematics underlying Maynard and Tao’s work, “but it was pretty much impossible for me,” he said. Their papers were far too complicated. Larsen tried to read related work, only to find it impenetrable as well. He kept at it, jumping from one result to another, until finally, in February 2021, he came across a paper he found both beautiful and comprehensible. Its subject: Carmichael numbers, those strange composite numbers that could sometimes pass themselves off as prime.

    All but Prime

    In the mid-17th century, the French mathematician Pierre de Fermat wrote a letter to his friend and confidant Frénicle de Bessy, in which he stated what would later be known as his “little theorem.” If N is a prime number, then b^N – b is always a multiple of N, no matter what b is. For instance, 7 is a prime number, and as a result, 2^7 – 2 (which equals 126) is a multiple of 7. Similarly, 3^7 – 3 is a multiple of 7, and so on.

    Mathematicians saw the potential for a perfect test of whether a given number is prime or composite. They knew that if N is prime, b^N – b is always a multiple of N. What if the reverse was also true? That is, if b^N – b is a multiple of N for all values of b, must N be prime?

    Alas, it turned out that in very rare cases, N can satisfy this condition and still be composite. The smallest such number is 561: For any integer b, b^561 – b is always a multiple of 561, even though 561 is not prime. Numbers like these were named after the mathematician Robert Carmichael, who is often credited with publishing the first example in 1910 (though the Czech mathematician Václav Šimerka independently discovered examples in 1885).

    Mathematicians wanted to better understand these numbers that so closely resemble the most fundamental objects in number theory, the primes. It turned out that in 1899—a decade before Carmichael’s result—another mathematician, Alwin Korselt, had come up with an equivalent definition. He simply hadn’t known if there were any numbers that fit the bill.

    According to Korselt’s criterion, a number N is a Carmichael number if and only if it satisfies three properties. First, it must have more than one prime factor. Second, no prime factor can repeat. And third, for every prime p that divides N, p – 1 also divides N – 1. Consider again the number 561. It’s equal to 3 × 11 × 17, so it clearly satisfies the first two properties in Korselt’s list. To show the last property, subtract 1 from each prime factor to get 2, 10 and 16. In addition, subtract 1 from 561. All three of the smaller numbers are divisors of 560. The number 561 is therefore a Carmichael number.

    Though mathematicians suspected that there are infinitely many Carmichael numbers, there are relatively few compared to the primes, which made them difficult to pin down. Then in 1994, Red Alford, Andrew Granville, and Carl Pomerance published a breakthrough paper [below] in which they finally proved that there are indeed infinitely many of these pseudoprimes.

    Unfortunately, the techniques they developed didn’t allow them to say anything about what those Carmichael numbers looked like. Did they appear in clusters along the number line, with large gaps in between? Or could you always find a Carmichael number in a short interval? “You’d think if you can prove there’s infinitely many of them,” Granville said, “surely you should be able to prove that there are no big gaps between them, that they should be relatively well spaced out.”

    In particular, he and his coauthors hoped to prove a statement that reflected this idea—that given a sufficiently large number X, there will always be a Carmichael number between X and 2X. “It’s another way of expressing how ubiquitous they are,” said Jon Grantham, a mathematician at the Institute for Defense Analyses who has done related work.

    But for decades, no one could prove it. The techniques developed by Alford, Granville and Pomerance “allowed us to show that there were going to be many Carmichael numbers,” Pomerance said, “but didn’t really allow us to have a whole lot of control about where they’d be.”

    Then, in November 2021, Granville opened up an email from Larsen, then 17 years old and in his senior year of high school. A paper [below] was attached—and to Granville’s surprise, it looked correct. “It wasn’t the easiest read ever,” he said. “But when I read it, it was quite clear that he wasn’t messing around. He had brilliant ideas.”

    Pomerance, who read a later version of the work, agreed. “His proof is really quite advanced,” he said. “It would be a paper that any mathematician would be really proud to have written. And here’s a high school kid writing it.”

    The key to Larsen’s proof was the work that had drawn him to Carmichael numbers in the first place: the results by Maynard and Tao on prime gaps.

    Unlikely—Not Impossible

    When Larsen first set out to show that you can always find a Carmichael number in a short interval, “it seemed that it was so obviously true, how hard can it be to prove?” he said. He quickly realized it could be very hard indeed. “This is a problem which tests the technology of our time,” he said.

    In their 1994 paper, Alford, Granville, and Pomerance had shown how to create infinitely many Carmichael numbers. But they hadn’t been able to control the size of the primes they used to construct them. That’s what Larsen would need to do to build Carmichael numbers that were relatively close in size. The difficulty of the problem worried his father, Michael Larsen. “I didn’t think it was impossible, but I thought it was unlikely he’d succeed,” he said. “I saw how much time he was spending on it … and I felt it would be devastating for him to give so much of himself to this and not get it.”

    Still, he knew better than to try to dissuade his son. “When Daniel commits to something that really interests him, he sticks with it through thick and thin,” he said.

    So Larsen returned to Maynard’s papers—in particular, to work showing that if you take certain sequences of enough numbers, some subset of those numbers must be prime. Larsen modified Maynard’s techniques to combine them with the methods used by Alford, Granville, and Pomerance. This allowed him to ensure that the primes he ended up with would vary in size—enough to produce Carmichael numbers that would fall within the intervals he wanted.

    “He has more control over things than we’ve ever had,” Granville said. And he achieved this through a particularly clever use of Maynard’s work. “It’s not easy … to use this progress on short gaps between primes,” said Kaisa Matomäki, a mathematician at the University of Turku in Finland. “It’s quite nice that he’s able to combine it with this question about the Carmichael numbers.”

    In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X. His proof works for much smaller intervals as well. Mathematicians now hope it will also help reveal other aspects of the behavior of these strange numbers. “It’s a different idea,” said Thomas Wright, a mathematician at Wofford College in South Carolina who works on pseudoprimes. “It changes a lot of things about how we might prove things about Carmichael numbers.”

    Grantham agreed. “Now you can do things you never thought of,” he said.

    Larsen, meanwhile, just started his freshman year at the Massachusetts Institute of Technology. He’s not sure what problem he might work on next, but he’s eager to learn what’s out there. “I’m just taking courses … and trying to be open-minded,” he said.

    “He did all this without an undergraduate education,” Grantham said. “I can only imagine what he’s going to be coming up with in graduate school.”

    Science papers:
    breakthrough paper 1994
    A paper 2021

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


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  • richardmitnick 12:11 pm on January 15, 2023 Permalink | Reply
    Tags: "Counting Curves", Mathematics, Professor of Mathematics Tony Yue Yu, The "non-Archimedean" world,   

    From The California Institute of Technology: “Counting Curves” 

    Caltech Logo

    From The California Institute of Technology

    Whitney Clavin
    (626) 395‑1944

    Professor of Mathematics Tony Yue Yu explains the mystery and power of the “non-Archimedean” world. Credit: Caltech.

    While counting is the first skill a child learns in mathematics, it’s also something studied at the highest levels of the discipline, albeit in a more exciting way. New professor of mathematics Tony Yue Yu research involves counting curves in a geometric space, which places his work in the field of enumerative geometry. One of the earliest examples of enumerative geometry is the Problem of Apollonius, named after a mathematician in ancient Greece.

    The Problem of Apollonius Credit: Wikipedia.

    In this problem, one counts the number of circles that are tangent to three given circles in a plane (black in the illustration). There are in general eight such tangent circles; one is shown in pink.

    Such questions are not only intuitively appealing, but also practically important because counting geometric objects with certain constraints is the same type of problem as counting the number of solutions to a system of equations.

    Yu is developing a new theory of curve counting via what is called non-Archimedean geometry. Typically, when you take two numbers such as 1 and 100, in which one number is less than the other, you can add the lesser number to itself again and again to eventually surpass the greater number (100). This concept stems from work by Archimedes, an ancient Greek mathematician. However, in non-Archimedean geometry, you can keep adding up smaller numbers but you will never surpass the larger number. Yu explains that these “exotic, non-intuitive” numbers are at the heart of his work.

    Yu, who grew up in Ningbo, China, completed his undergraduate studies in mathematics at Peking University in 2010, and later completed his graduate studies at the Ecole normale supérieure in Paris. He was a permanent researcher in the French National Centre for Scientific Research until joining the Caltech faculty in 2021.

    We met with Yu over Zoom to learn more about his research and how it relates to a concept familiar to physicists known as mirror symmetry.
    When did you first become interested in math?

    I have been fascinated by math and science since childhood. Growing up in Ningbo, there were plenty of science activities and competitions for kids, and I enjoyed all of them. I went to Peking University, majoring in mathematics, because in high school I was able to read university textbooks on many other science subjects but couldn’t understand much from the math textbooks. I became very curious about modern mathematics.

    Then I went to Ecole normale supérieure in Paris for graduate school. Paris is the birthplace of modern algebraic geometry, founded by Alexander Grothendieck in the 1960s. People like to say that Paris is the center of the fashion world, but it is also a center of mathematical research in many areas. Once in Paris, surrounded by so many mathematicians, it was natural for me to pursue mathematical research.

    Can you tell us more about non-Archimedean geometry?

    The Archimedean property says that given any two positive numbers A < B, if we add A to itself sufficiently many times, the sum A+ ⋯ +A will eventually exceed B.

    The Archimedean property Credit: Wikipedia.

    You will say that this is obvious because this is how length behaves in our daily life. However, in modern mathematics, there is a great interest to study quantities and geometric spaces where the Archimedean property fails. We call them non-Archimedean numbers and non-Archimedean spaces. In this realm, numbers do not fall on a number line nor represent any notion of distance. They are exotic and don’t match our intuition.

    Non-Archimedean geometry is a branch of algebraic geometry, where we study geometric shapes defined over non-Archimedean numbers. Since we do not live in a non-Archimedean world, it is hard to study non-Archimedean spaces, and many research mathematicians considered it to be a difficult and abstract field.

    I became interested in the field while I was a graduate student in Paris. One day I asked my advisor what a non-Archimedean space is. He replied that it is some very “hairy” space. I used to think of mathematical objects as austere and solemn, and I couldn’t believe how a geometric space can be hairy like an animal! I became very fascinated by the subject afterward.

    Can you tell us more about enumerative geometry?

    Enumerative geometry is about counting geometric objects, such as the Problem of Apollonius, in which one counts circles in a plane. While it is fun and important to count and compute the precise numbers, the thrill of the field is to discover deeper structural relations behind these numbers. One of the most mysterious relations is described by so-called mirror symmetry, which is a duality of shapes first discovered by theoretical physicists studying string theory, a mathematical theory that aims to describe the fundamental particles and forces in nature.

    Regardless of whether string theory can be proven by experiments, it has made a great impact on mathematical research. In mirror symmetry, the numbers of curves in one space can be related to solutions of differential equations on the mirror space. The full extent of this phenomenon, as well as its underlying mathematical mechanism, are still largely unknown.

    What problems are you working on?

    I initiated the study of enumerative geometry using non-Archimedean methods, in particular with the aim of solving conjectures in the field of mirror symmetry. In fact, non-Archimedean spaces appear naturally in the study of mirror symmetry via a process known as degeneration of spaces. One can think of degeneration as crashing a big complicated space into smaller, simpler broken pieces. The parameter for the degeneration becomes a non-Archimedean number, and the degeneration process gives rise to a non-Archimedean space. However, most researchers were not keen on applying non-Archimedean methods to curve counting problems because non-Archimedean geometry was considered to be exotic and difficult. My research has been exploring this direction in the last few years, and it has turned out to be a rewarding experience.

    Counting non-Archimedean curves in mirror symmetry. Credit: Tony Yue Yu.

    I aim to further develop this non-Archimedean approach and hopefully make new contributions to the mathematical foundation of mirror symmetry. It is also important to compare this work with methods studied by other researchers. While mirror symmetry completely revolutionized the field of enumerative geometry, I also look forward to exploring applications of mirror symmetry to broader areas of algebraic geometry, such as moduli theory and birational geometry. Both concern the classification of spaces, and classification has always been a central theme across different areas of mathematics.

    Is there anything else you would like to add?

    In addition to being the indispensable tool for science and technology, math is also an art. Sometimes you hear a piece of music and you like it. Math has a lot of surprises but requires years of training to fully appreciate.

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    The California Institute of Technology is a private research university in Pasadena, California. The university is known for its strength in science and engineering, and is one among a small group of institutes of technology in the United States which is primarily devoted to the instruction of pure and applied sciences.

    The California Institute of Technology was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began attracting influential scientists such as George Ellery Hale, Arthur Amos Noyes, and Robert Andrews Millikan in the early 20th century. The vocational and preparatory schools were disbanded and spun off in 1910 and the college assumed its present name in 1920. In 1934, The California Institute of Technology was elected to the Association of American Universities, and the antecedents of National Aeronautics and Space Administration ‘s Jet Propulsion Laboratory, which The California Institute of Technology continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán.

    The California Institute of Technology has six academic divisions with strong emphasis on science and engineering. Its 124-acre (50 ha) primary campus is located approximately 11 mi (18 km) northeast of downtown Los Angeles. First-year students are required to live on campus, and 95% of undergraduates remain in the on-campus House System at The California Institute of Technology. Although The California Institute of Technology has a strong tradition of practical jokes and pranks, student life is governed by an honor code which allows faculty to assign take-home examinations. The The California Institute of Technology Beavers compete in 13 intercollegiate sports in the NCAA Division III’s Southern California Intercollegiate Athletic Conference (SCIAC).

    As of October 2020, there are 76 Nobel laureates who have been affiliated with The California Institute of Technology, including 40 alumni and faculty members (41 prizes, with chemist Linus Pauling being the only individual in history to win two unshared prizes). In addition, 4 Fields Medalists and 6 Turing Award winners have been affiliated with The California Institute of Technology. There are 8 Crafoord Laureates and 56 non-emeritus faculty members (as well as many emeritus faculty members) who have been elected to one of the United States National Academies. Four Chief Scientists of the U.S. Air Force and 71 have won the United States National Medal of Science or Technology. Numerous faculty members are associated with the Howard Hughes Medical Institute as well as National Aeronautics and Space Administration. According to a 2015 Pomona College study, The California Institute of Technology ranked number one in the U.S. for the percentage of its graduates who go on to earn a PhD.


    The California Institute of Technology is classified among “R1: Doctoral Universities – Very High Research Activity”. Caltech was elected to The Association of American Universities in 1934 and remains a research university with “very high” research activity, primarily in STEM fields. The largest federal agencies contributing to research are National Aeronautics and Space Administration; National Science Foundation; Department of Health and Human Services; Department of Defense, and Department of Energy.

    In 2005, The California Institute of Technology had 739,000 square feet (68,700 m^2) dedicated to research: 330,000 square feet (30,700 m^2) to physical sciences, 163,000 square feet (15,100 m^2) to engineering, and 160,000 square feet (14,900 m^2) to biological sciences.

    In addition to managing NASA-JPL/Caltech , The California Institute of Technology also operates the Caltech Palomar Observatory; The Owens Valley Radio Observatory;the Caltech Submillimeter Observatory; the W. M. Keck Observatory at the Mauna Kea Observatory; the Laser Interferometer Gravitational-Wave Observatory at Livingston, Louisiana and Hanford, Washington; and Kerckhoff Marine Laboratory in Corona del Mar, California. The Institute launched the Kavli Nanoscience Institute at The California Institute of Technology in 2006; the Keck Institute for Space Studies in 2008; and is also the current home for the Einstein Papers Project. The Spitzer Science Center, part of the Infrared Processing and Analysis Center located on The California Institute of Technology campus, is the data analysis and community support center for NASA’s Spitzer Infrared Space Telescope [no longer in service].

    The California Institute of Technology partnered with University of California at Los Angeles to establish a Joint Center for Translational Medicine (UCLA-Caltech JCTM), which conducts experimental research into clinical applications, including the diagnosis and treatment of diseases such as cancer.

    The California Institute of Technology operates several Total Carbon Column Observing Network stations as part of an international collaborative effort of measuring greenhouse gases globally. One station is on campus.

  • richardmitnick 7:57 am on January 12, 2023 Permalink | Reply
    Tags: "Preparing for a changing climate", A multi-institutional effort to identify the best models to calculate flood risk at coastal military installations where climate change threatens to increase the risk of flood damage., , , , , , Many military installations are located along the coast and they can’t be easily relocated. They need to be protected., Mathematics, , The findings could have broader implications for coastal communities., The goal is to be able to accurately predict what kind of flooding or damage a certain site might experience during a hurricane impact., The models have to be able to process information quickly enough so that there’s time for a response., The more complex the model is the more physics it includes and the more computationally demanding it is., , UD civil engineers lead research to examine models for coastal readiness at U.S. military bases.   

    From The University of Delaware : “Preparing for a changing climate” 

    U Delaware bloc

    From The University of Delaware

    Maddy Lauria
    Photo courtesy of Christopher Lashley, Stephanie Patch and NASA.
    Photo illustrations by Joy Smoker.

    Jack Puleo, chair of the University of Delaware’s Department of Civil and Environmental Engineering, is leading a research effort that could have broad implications for coastal communities and calculating risk in the face of a changing climate and rising sea levels.

    UD civil engineers lead research to examine models for coastal readiness at U.S. military bases.

    University of Delaware civil engineers are leading a multi-institutional effort to identify the best models to calculate flood risk at coastal military installations where climate change threatens to increase the risk of flood damage from sea level rise and storm surge.

    The four-year project, which launched in mid-2022 and will run through spring 2025, is funded by a $2.2 million grant from the U.S. Department of Defense (DoD). Project partners include faculty and students from the Netherlands, North Carolina State University, the University of South Alabama, Texas A&M and the United States Geological Survey (USGS).

    “Many military installations are located along the coast and they can’t be easily relocated. They need to be protected,” said Jack Puleo, chair of UD’s Department of Civil and Environmental Engineering and project lead. “To do that, we need to understand what the flooding risk is.”

    The DoD-funded research will explore numerical models that calculate total water levels in the face of sea level rise, tides, wind-induced surge, waves and other environmental variables to determine which approaches not only perform the best but are also the most cost-effective. The team of researchers will apply their work to three military sites: the Virginia-based Naval Station Norfolk on the Atlantic Coast, Tyndall Air Force Base on the Gulf Coast of Florida and the Ronald Reagan Ballistic Missile Defense Test Site on the Marshall Islands in the Pacific Ocean.

    The goal is to be able to accurately predict what kind of flooding or damage a certain site might experience during a hurricane impact, for example, when there’s been another foot of sea level rise.

    “But it’s not just getting wet that’s important,” Puleo said. “It’s about flooding duration and depth. If a prediction says there will be 1 inch of water on a roadway, maybe you don’t care as much. But if it says you’ll have 1 foot of water for multiple tidal cycles, that’s important to know. It could hamper critical services and evacuation.”

    This image shows one of the areas a team of civil engineers will be focusing on — the coastline near Tyndall Air Force Base along the Gulf of Mexico — during a multi-year research project to examine the varying strengths and weaknesses of coastal flooding models, particularly in the face of changing water levels.

    Their findings could have broader implications for coastal communities by identifying which applications work best in which settings because running high-fidelity models isn’t cheap or easy.

    Modeling strengths and weaknesses

    There is a wide range of predictive models available to use, from those that handle basic calculations (but are still highly technical) to those that can produce highly localized results. Combining those models with witness accounts and existing data will help researchers “tease out the importance of knowing the fine details,” Puleo said.

    The question is how much information is really needed to make accurate predictions that could help these military installations become more resilient in the face of a changing climate, especially along the coast. It’s also about timing: the models have to be able to process information quickly enough so that there’s time for a response, such as moving assets out of the way if necessary.

    “We’re the team testing out all of these models and methods to be able to provide a kind of roadmap for when to use which model and what it will cost computationally or resource-wise to be able to do that,” said Stephanie Patch, an associate professor at the University of South Alabama’s Department of Civil, Coastal, and Environmental Engineering. The answer would largely depend on the event — a heavy rain or a major hurricane — as well as the specific location.

    The military is interested in learning about the best options because there can be a steep cost associated with running the higher-end models — upwards of $250,000 per site for data collection, supercomputer access and manpower to generate model input, at a rough estimate. On the other hand, there could also be a steep cost with responding to an event that never happens if the model’s prediction doesn’t play out — or the opposite if an event turns unexpectedly catastrophic and there’s no time to respond.

    These images show water elevations after Hurricane Michael of 2018 at Tyndall Air Force Base along the Gulf of Mexico. Total water levels were estimated using a model called XBeach, run by the University of South Alabama’s Stephanie Patch and colleagues.

    While Patch is focusing on a model that’s very closely tied to a small area of beach and dune and the impacts of erosion, North Carolina State University’s Casey Dietrich is working with larger-scale models capable of simulating storm effects over large areas, like an entire state or the entire Gulf of Mexico. But the information from the varying models can be linked to help the smaller-scale studies make more accurate predictions, Dietrich explained.

    “The goal is to provide guidance to the DoD about the strengths and weaknesses of each model in comparison. They’re all going to have things they’re good with and things they struggle with,” Dietrich said. Those comparisons will help the agencies decide what types of models they want to use to get what types of information — depending on how much time, effort and funding they want to commit.

    There’s also a goal of reducing cost and building smarter models, he said.

    “If we are able to improve our predictions at very specific sites along the coast, we also can have better predictions at other specific sites along the coast, like someone’s house or a bridge or other infrastructure,” Dietrich said.

    Still, differences in the geographic location of the military facilities themselves will play a role in the physics of varying environmental factors, such as wind-driven waves or storm surge, and how those variables interact with the land. That’s why researchers are exploring sites on the Atlantic, Gulf and Pacific coasts. 

    But knowing everything everywhere isn’t always possible, Puleo said. Information on what the seafloor or topography looks like may rely on data collected decades ago or sparse patches of information.

    “There’s so many models to choose, and they’re not all easy to just pick up and use,” said Patch. “I think this project is so great because we’re getting a team of people together who have this expertise in different models who can determine the benefits of those.”

    Planning for the future

    Making as-accurate-as-possible predictions despite any data gaps and potential funding restraints is part of the real-world balance decision-makers must tackle in the face of storm preparedness. 

    UD postdoc Christopher Lashley is using data from 2011’s Hurricane Irene to see how a particular level of modeling will perform. His job, he said, is to make sure he’s giving the model the correct input — because the model is only as accurate as the information it’s given.

    “The more complex the model is, the more physics it includes, the more computationally demanding it is,” Lashley said. “One simulation could take maybe 100 people with individual laptops running at the same time, if you weren’t using a supercomputer. Lesser models can be run in one hour or a few minutes. It can vary significantly.”

    These three images show how the position of a hurricane can impact the water level, due to surge, in certain coastal areas.

    UD Professor Fengyan Shi, a numerical modeling expert with decades of experience and a core faculty member of UD’s Center for Applied Coastal Research, will lead the modeling group. He said working with fluid environments is very complicated because of the various elements you have to consider, like wind fields and how waves are generated.

    Add on top of that the long-term process of sea level rise and physics happening in different places, such as the way water flows in a harbor versus the open ocean, and it’s easy to see how researchers can become very detailed with their modeling approaches.

    “This is real applied research,” Shi said, noting that it will also help researchers further study the impact of physics in model predictions.

    Ultimately, what the team tests and validates could be useful to everyone, Lashley said. Especially if findings indicate that the less-complex models work well at predicting, say, devastating impacts from a hurricane that would require evacuations days in advance, the coastal engineering work they’re doing could ultimately benefit countries and communities without the access to supercomputers or time to wait for slower models to be run.

    “If you know, then you can plan,” he said.

    This forward-looking kind of research is also what lies ahead in the future of coastal engineering, said Patch.

    “We’re learning a lot about the models in terms of how they compare with each other,” she said. “I hope the outcome of this work doesn’t just benefit these specific locations, but also military installations around the world and communities around the world. It’s so translatable and transferable. I would love to see an outcome of this project — even if it’s indirect — to learn enough to apply it worldwide on all of our coasts as climate changes.”

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


    Please help promote STEM in your local schools.

    Stem Education Coalition

    U Delaware campus

    The University of Delaware is a public land-grant research university located in Newark, Delaware. University of Delaware (US) is the largest university in Delaware. It offers three associate’s programs, 148 bachelor’s programs, 121 master’s programs (with 13 joint degrees), and 55 doctoral programs across its eight colleges. The main campus is in Newark, with satellite campuses in Dover, the Wilmington area, Lewes, and Georgetown. It is considered a large institution with approximately 18,200 undergraduate and 4,200 graduate students. It is a privately governed university which receives public funding for being a land-grant, sea-grant, and space-grant state-supported research institution.

    The University of Delaware is classified among “R1: Doctoral Universities – Very high research activity”. According to The National Science Foundation, UD spent $186 million on research and development in 2018, ranking it 119th in the nation. It is recognized with the Community Engagement Classification by the Carnegie Foundation for the Advancement of Teaching.

    The University of Delaware is one of only four schools in North America with a major in art conservation. In 1923, it was the first American university to offer a study-abroad program.

    The University of Delaware traces its origins to a “Free School,” founded in New London, Pennsylvania in 1743. The school moved to Newark, Delaware by 1765, becoming the Newark Academy. The academy trustees secured a charter for Newark College in 1833 and the academy became part of the college, which changed its name to Delaware College in 1843. While it is not considered one of the colonial colleges because it was not a chartered institution of higher education during the colonial era, its original class of ten students included George Read, Thomas McKean, and James Smith, all three of whom went on to sign the Declaration of Independence. Read also later signed the United States Constitution.

    Science, Technology and Advanced Research (STAR) Campus

    On October 23, 2009, The University of Delaware signed an agreement with Chrysler to purchase a shuttered vehicle assembly plant adjacent to the university for $24.25 million as part of Chrysler’s bankruptcy restructuring plan. The university has developed the 272-acre (1.10 km^2) site into the Science, Technology and Advanced Research (STAR) Campus. The site is the new home of University of Delaware (US)’s College of Health Sciences, which includes teaching and research laboratories and several public health clinics. The STAR Campus also includes research facilities for University of Delaware (US)’s vehicle-to-grid technology, as well as Delaware Technology Park, SevOne, CareNow, Independent Prosthetics and Orthotics, and the East Coast headquarters of Bloom Energy. In 2020 [needs an update], University of Delaware expects to open the Ammon Pinozzotto Biopharmaceutical Innovation Center, which will become the new home of the UD-led National Institute for Innovation in Manufacturing Biopharmaceuticals. Also, Chemours recently opened its global research and development facility, known as the Discovery Hub, on the STAR Campus in 2020. The new Newark Regional Transportation Center on the STAR Campus will serve passengers of Amtrak and regional rail.


    The university is organized into nine colleges:

    Alfred Lerner College of Business and Economics
    College of Agriculture and Natural Resources
    College of Arts and Sciences
    College of Earth, Ocean and Environment
    College of Education and Human Development
    College of Engineering
    College of Health Sciences
    Graduate College
    Honors College

    There are also five schools:

    Joseph R. Biden, Jr. School of Public Policy and Administration (part of the College of Arts & Sciences)
    School of Education (part of the College of Education & Human Development)
    School of Marine Science and Policy (part of the College of Earth, Ocean and Environment)
    School of Nursing (part of the College of Health Sciences)
    School of Music (part of the College of Arts & Sciences)

  • richardmitnick 9:22 am on December 26, 2022 Permalink | Reply
    Tags: "A New Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations", "Proof by Computer", A 177-page proof — the result of a decade-long research program — makes significant use of computers., A million-dollar Millennium Prize awaits anyone who can prove whether similar failures occur in the Navier-Stokes equations-a generalization of the Euler equations that accounts for viscosity., Euler equations, For centuries mathematicians have sought to understand and model the motion of fluids., In fluid mechanics computer-assisted proofs are still a relatively new technique., It gets even more complicated if you’re trying to model a fluid that has viscosity-as almost all real-world fluids do., It’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity but it can’t actually reach it, Mathematicians want to establish whether equations that model fluid flow can sometimes fail or “blow up.”, Mathematics, Navier-Stokes equations, The "Move to Self-Similar Land", The motion of fluids, These efforts mark a growing trend in the field of fluid dynamics: the use of computers to solve important problems.,   

    From “Quanta Magazine” : “A New Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations” 

    From “Quanta Magazine”

    11.16.22 [Just found this via Wired
    Jordana Cepelewicz

    Mathematicians want to establish whether equations that model fluid flow can sometimes fail, or “blow up.” Quanta Magazine

    Mathematicians want to establish whether equations that model fluid flow can sometimes fail, or “blow up.” Credit: DVDP for Quanta Magazine.

    For centuries mathematicians have sought to understand and model the motion of fluids. The equations that describe how ripples crease the surface of a pond have also helped researchers to predict the weather, design better airplanes, and characterize how blood flows through the circulatory system. These equations are deceptively simple when written in the right mathematical language. However, their solutions are so complex that making sense of even basic questions about them can be prohibitively difficult.

    Perhaps the oldest and most prominent of these equations, formulated by Leonhard Euler more than 250 years ago, describe the flow of an ideal, incompressible fluid: a fluid with no viscosity, or internal friction, that cannot be forced into a smaller volume. “Almost all nonlinear fluid equations are kind of derived from the Euler equations,” said Tarek Elgindi, a mathematician at Duke University. “They’re the first ones, you could say.”

    Yet much remains unknown about the Euler equations — including whether they’re always an accurate model of ideal fluid flow. One of the central problems in fluid dynamics is to figure out if the equations ever fail, outputting nonsensical values that render them unable to predict a fluid’s future states.

    Mathematicians have long suspected that there exist initial conditions that cause the equations to break down. But they haven’t been able to prove it.

    In a new science paper [below] posted online last month, a pair of mathematicians has shown that a particular version of the Euler equations does indeed sometimes fail. The proof marks a major breakthrough — and while it doesn’t completely solve the problem for the more general version of the equations, it offers hope that such a solution is finally within reach. “It’s an amazing result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”

    There’s just one catch.

    The 177-page proof — the result of a decade-long research program — makes significant use of computers. This arguably makes it difficult for other mathematicians to verify it. (In fact, they are still in the process of doing so, though many experts believe the new work will turn out to be correct.) It also forces them to reckon with philosophical questions about what a “proof” is, and what it will mean if the only viable way to solve such important questions going forward is with the help of computers.

    Sighting the Beast

    In principle, if you know the location and velocity of each particle in a fluid, the Euler equations should be able to predict how the fluid will evolve for all time. But mathematicians want to know if that’s actually the case. Perhaps in some situations, the equations will proceed as expected, producing precise values for the state of the fluid at any given moment, only for one of those values to suddenly skyrocket to infinity. At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”

    Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow. But “as of a few years ago, what people were able to do fell very, very far short of [proving blowup],” said Charlie Fefferman, a mathematician at Princeton University.

    It gets even more complicated if you’re trying to model a fluid that has viscosity (as almost all real-world fluids do). A million-dollar Millennium Prize from the Clay Mathematics Institute awaits anyone who can prove whether similar failures occur in the Navier-Stokes equations, a generalization of the Euler equations that accounts for viscosity.

    In 2013, Thomas Hou, a mathematician at the California Institute of Technology, and Guo Luo, now at the Hang Seng University of Hong Kong, proposed a scenario in which the Euler equations would lead to a singularity. They developed a computer simulation of a fluid in a cylinder whose top half swirled clockwise while its bottom half swirled counterclockwise. As they ran the simulation, more complicated currents started to move up and down. That, in turn, led to strange behavior along the boundary of the cylinder where opposing flows met. The fluid’s vorticity — a measure of rotation — grew so fast that it seemed poised to blow up.

    Merrill Sherman/Quanta Magazine.

    Hou and Luo’s work was suggestive, but not a true proof. That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it — meaning that the solution might be very accurate, but it’s still an approximation. Without the backing of a mathematical proof, the value of the vorticity might only seem to be increasing to infinity because of some artifact of the simulation. The solutions might instead grow to enormous numbers before again subsiding.

    Such reversals had happened before: A simulation would indicate that a value in the equations blew up, only for more sophisticated computational methods to show otherwise. “These problems are so delicate that the road is littered with the wreckage of previous simulations,” Fefferman said. In fact, that’s how Hou got his start in this area: Several of his earlier results disproved the formation of hypothetical singularities.

    Still, when he and Luo published their solution, most mathematicians thought it was very likely a true singularity. “It was very meticulous, very precise,” said Vladimir Sverak, a mathematician at the University of Minnesota. “They really went to great lengths to establish that this is a real scenario.” Subsequent work by Elgindi, Sverak and others only strengthened that conviction.

    But a proof was elusive. “You’ve sighted the beast,” Fefferman said. “Then you try to capture it.” That meant showing that the approximate solution that Hou and Luo so carefully simulated is, in a specific mathematical sense, very, very close to an exact solution of the equations.

    Now, nine years after that first sighting, Hou and his former graduate student Jiajie Chen have finally succeeded in proving the existence of that nearby singularity.

    The Move to Self-Similar Land

    Hou, later joined by Chen, took advantage of the fact that, upon closer analysis, the approximate solution from 2013 seemed to have a special structure. As the equations evolved through time, the solution displayed what’s called a self-similar pattern: Its shape later on looked a lot like its earlier shape, only re-scaled in a specific way.

    As a result, the mathematicians didn’t need to try to look at the singularity itself. Instead, they could study it indirectly by focusing on an earlier point in time. By zooming in on that part of the solution at the right rate — determined based on the solution’s self-similar structure — they could model what would happen later on, including at the singularity itself.

    It took a few years for them to find a self-similar analogue to the 2013 blowup scenario. (Earlier this year, another team of mathematicians, which included Buckmaster, used different methods to find a similar approximate solution. They are currently using that solution to develop an independent proof of singularity formation.)

    With an approximate self-similar solution in hand, Hou and Chen needed to show that an exact solution exists nearby. Mathematically, this is equivalent to proving that their approximate self-similar solution is stable — that even if you were to slightly perturb it and then evolve the equations starting at those perturbed values, there’d be no way to escape a small neighborhood around the approximate solution. “It’s like a black hole,” Hou said. “If you start with a profile close by, you’ll be sucked in.”

    But having a general strategy was just one step toward the solution. “Fussy details matter,” Fefferman said. As Hou and Chen spent the next several years working out those details, they found that they had to rely on computers once again — but this time in an entirely new way.

    A Hybrid Approach

    Among their first challenges was figuring out the exact statement they had to prove. They wanted to show that if they took any set of values close to their approximate solution and plugged it into the equations, the output wouldn’t be able to stray far. But what does it mean for an input to be “close” to the approximate solution? They had to specify this in a mathematical statement — but there are many ways to define the notion of distance in this context. For their proof to work, they needed to choose the correct one.

    “It has to measure different physical effects,” said Rafael de la Llave, a mathematician at the Georgia Institute of Technology. “So it needs to be chosen using a deep understanding of the problem.”

    Once they had the right way to describe “closeness,” Hou and Chen had to prove the statement, which boiled down to a complicated inequality involving terms from both the re-scaled equations and the approximate solution. The mathematicians had to make sure that the values of all those terms balanced out to something very small: If one value ended up being large, other values had to be negative or kept in check.

    “If you make something a little too big or a little too small, the whole thing breaks down,” said Javier Gómez-Serrano, a mathematician at Brown University. “So it’s very, very careful, delicate work.”

    “It’s a really fierce fight,” Elgindi added.

    To get the tight bounds they needed on all these different terms, Hou and Chen broke the inequality into two major parts. They could take care of the first part by hand, with techniques including one that dates back to the 18th century, when the French mathematician Gaspard Monge sought an optimal way of transporting soil to build fortifications for Napoleon’s army. “Stuff like this has been done before, but I found it striking that [Hou and Chen] used it for this,” Fefferman said.

    That left the second part of the inequality. Tackling it would require computer assistance. For starters, there were so many calculations that needed to be done, and so much precision required, that “the amount of work you’d have to do with pencil and paper would be staggering,” de la Llave said. To get various terms to balance out, the mathematicians had to perform a series of optimization problems that are relatively easy for computers but exceedingly time-consuming for humans. Some of the values also depended on quantities from the approximate solution; since that was calculated using a computer, it was more straightforward to also use a computer to perform these additional computations.

    “If you try to manually do some of these estimates, you’re probably going to overestimate at some point, and then you lose,” said Gómez-Serrano. “The numbers are so tiny and tight … and the margin is incredibly thin.”

    But because computers can’t manipulate an infinite number of digits, tiny errors inevitably occur. Hou and Chen had to carefully track those errors, to make sure they didn’t interfere with the rest of the balancing act.

    Ultimately, they were able to find bounds for all the terms, completing the proof: The equations had indeed produced a singularity.

    Proof by Computer

    It remains open whether more complicated equations — the Euler equations without the presence of a cylindrical boundary and the Navier-Stokes equations — can develop a singularity. “But [this work] at least gives me hope,” Hou said. “I see a path forward, a way to maybe even eventually resolve the full Millennium problem.”

    Meanwhile, Buckmaster and Gómez-Serrano are working on a computer-assisted proof of their own — one they hope will be more general, and therefore capable of tackling not just the problem that Hou and Chen solved, but also scores of others.

    These efforts mark a growing trend in the field of fluid dynamics: the use of computers to solve important problems.

    “In a number of different areas of mathematics, it’s occurring more and more frequently,” said Susan Friedlander, a mathematician at the University of Southern California.

    But in fluid mechanics computer-assisted proofs are still a relatively new technique. In fact, when it comes to statements about singularity formation, Hou and Chen’s proof is the first of its kind: Previous computer-assisted proofs were only able to tackle toy problems in the area.

    Such proofs aren’t so much controversial as “a matter of taste,” said Peter Constantin of Princeton University. Mathematicians generally agree that a proof has to convince other mathematicians that some line of reasoning is correct. But, many argue, it should also improve their understanding of why a particular statement is true, rather than simply provide validation that it’s correct. “Do we learn anything fundamentally new, or do we just know the answer to the question?” Elgindi said. “If you view mathematics as an art, then this is not so aesthetically pleasing.”

    “A computer can help. It’s wonderful. It gives me insight. But it doesn’t give me a full understanding,” Constantin added. “Understanding comes from us.”

    For his part, Elgindi still hopes to work out an alternative proof of blowup entirely by hand. “I’m overall happy this exists,” he said of Hou and Chen’s work. “But I take it as more of a motivation to try to do it in a less computer-dependent way.”

    Other mathematicians view computers as a vital new tool that will make it possible to attack previously intractable problems. “Now the work is no longer just paper and pencil,” Chen said. “You have the option of using something more powerful.”

    According to him and others (including Elgindi, despite his personal preference for writing proofs by hand), there’s a good possibility that the only way to solve big problems in fluid dynamics — that is, problems that involve increasingly complicated equations — might be to rely heavily on computer assistance. “It looks to me as if trying to do this without making heavy use of computer-assisted proofs is like tying one or possibly two hands behind your back,” Fefferman said.

    If that does end up being the case and “you don’t have any choice,” Elgindi said, “then people … such as myself, who would say that this is suboptimal, should be quiet.” That would also mean that more mathematicians would need to start learning the skills needed to write computer-assisted proofs — something that Hou and Chen’s work will hopefully inspire. “I think there were a lot of people who were simply waiting for someone to solve such a problem before investing any of their own time into this approach,” Buckmaster said.

    That said, when it comes to debates about the extent to which mathematicians should rely on computers, “it’s not that you need to pick a side,” Gómez-Serrano said. “[Hou and Chen’s] proof wouldn’t work without the analysis, and the proof wouldn’t work without the computer assistance. … I think the value is that people can speak the two languages.”

    With that, de la Llave said, “there’s a new game in town.”

    Science paper:
    new science paper
    See the science paper for instructive material with images.

    See the full article here .

    Comments are invited and will be appreciated, especially if the reader finds any errors which I can correct. Use “Reply”.


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    Formerly known as Simons Science News, Quanta Magazine is an editorially independent online publication launched by the Simons Foundation to enhance public understanding of science. Why Quanta? Albert Einstein called photons “quanta of light.” Our goal is to “illuminate science.” At Quanta Magazine, scientific accuracy is every bit as important as telling a good story. All of our articles are meticulously researched, reported, edited, copy-edited and fact-checked.

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