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  • richardmitnick 11:24 am on May 28, 2017 Permalink | Reply
    Tags: , , , , , , Mathematics, ,   

    From LLNL: Women in STEM-“Lab engages girls at San Joaquin STEM event” 

    Lawrence Livermore National Laboratory

    Carenda L Martin

    The Laboratory participated in an educational outreach event held last month titled, “Engaging Girls in STEM: Making a Connection for Action,” at the San Joaquin County Office of Education facility in Stockton.

    More than 300 young women in grades 6-12 attended the program, which is part of a statewide initiative to encourage young girls and women to pursue education and careers in science, technology, engineering and math (STEM) related fields. The event was hosted by the San Joaquin County Office of Education, State Department of Education and the California Commission on the Status of Women and Girls.

    Girls donned 3D googles to take a 360-degree virtual reality tour of the Lab’s National Ignition and Additive Manufacturing facilities.
    No image credit.

    A panel of women working in STEM fields was featured along with an exhibitor fair, showcasing various STEM programs and professions, such as LLNL, Association of Women in Science, CSU Sacramento, San Joaquin Delta College, University of the Pacific, Stockton Astronomical Society and the World of Wonders (WOW) Museum. Occupational therapists, engineers, microbiologists, neuroscientists, physicians and computer scientists also showcased hands-on, industry-based activities.

    The Laboratory was well represented with a booth that featured 360 degree tours of the National Ignition and Additive Manufacturing facilities via 3D goggles, and a booth with giveaways and information about the San Joaquin Expanding Your Horizons conference for girls, which is now in its 25th year and led and organized by a committee of Lab volunteers.

    Also featured was the Laboratory’s popular Fun With Science program, presented by Nick Williams, featuring experiments involving states of matter, chemistry, electricity, air pressure, etc.

    Employee volunteers included Cary Gellner, Carrie Martin, Norma McTyer (retired), Jeene Villanueva along with Joanna Albala, LLNL’s education program manager, who facilitated the Lab’s involvement.

    STEM Lab volunteers included (from left) Joanna Albala, Jeene Villanueva, Cary Gellner, Carrie Martin, Norma McTyer (retired) and Nick Williams.

    See the full article here .

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    LLNL Campus

    Operated by Lawrence Livermore National Security, LLC, for the Department of Energy’s National Nuclear Security
    DOE Seal

  • richardmitnick 8:43 pm on May 21, 2017 Permalink | Reply
    Tags: , , , , , Mathematics, NanoFab, ,   

    From NIST: “Nanocollaboration Leads to Big Things” 


    May 12, 2017 [Nothing like being timely getting into social media.]

    Ben Stein
    (301) 975-2763

    Entrance to NIST’s Advanced Measurement Laboratory in Gaithersburg, Maryland. Credit: Photo Courtesy HDR Architecture, Inc./Steve Hall Copyright Hedrich Blessing

    Roche Sequencing Solutions engineer Juraj Topolancik was looking for a way to decode DNA from cancer patients in a matter of minutes.

    Rajesh Krishnamurthy, a researcher with the startup company 3i Diagnostics, needed help in fabricating a key component of a device that rapidly identifies infection-causing bacteria.

    Ranbir Singh, an engineer with GeneSiC Semiconductor Inc., in Dulles, Virginia, sought to construct and analyze a semiconductor chip that transmits voltages large enough to power electric cars and spacecraft.

    These researchers all credit the NanoFab, located at the Center for Nanoscale Science and Technology (CNST) on the Gaithersburg, Maryland campus of the National Institute of Standards and Technology (NIST). The NanoFab provides cutting-edge nanotechnology capabilities for NIST scientists that is also accessible to outside users, with supplying the state-of-art tools, know-how and dependability to realize their goals.

    Learn more about the CNST NanoFab, where scientists from government, academia and industry can use commercial, state-of-the-art tools at economical rates, and get help from dedicated, full-time technical support staff. Voices: David Baldwin (Great Ball of Light, Inc.), Elisa Williams (Scientific & Biomedical Microsystems), George Coles (Johns Hopkins Applied Physics Laboratory) and William Osborn (NIST).

    When Krishnamurthy, whose company is based in Germantown, Maryland, needed an infrared filter for the bacteria-identifying chip, proximity was but one factor in reaching out to the NanoFab.

    “Even more important was the level of expertise you have here,” he says. “The attention to detail and the trust we have in the staff is so important—we didn’t have to worry if they would do a good job, which gives us tremendous peace of mind,” Krishnamurthy notes.

    The NanoFab also aided his project in another, unexpected way. Krishnamurthy had initially thought that the design for his company’s device would require a costly, highly customized silicon chip. But in reviewing design plans with engineers at the NanoFab, “they came up with a very creative way” to use a more standard, less expensive silicon wafer that would achieve the same goals, he notes.

    “The impact in the short term is that we didn’t have to pay as much [to build and test] the device at the NanoFab, which matters quite a bit because we’re a start-up company,” says Krishnamurthy. “In the long run, this will be a huge factor in [enabling us to mass produce] the device, keeping our costs low because, thanks to the input from the NanoFab, the source material is not a custom material.”

    Singh came to the NanoFab with a different mission. His company is developing a gallium nitride semiconductor device durable enough to transmit hundreds to thousands of volts without deteriorating. He relies on the NanoFab’s metal deposition tools and high-resolution lithography instruments to finish building and assess the properties of the device.

    Semiconductor device, fabricated with the help of the NanoFab, designed to transmit high voltages.
    Credit: GeneSiC Semiconductor Inc.

    “Not only is there a wide diversity of tools, but within each task there are multiple technologies,” Singh adds.

    For instance, he notes, technologies offered at the NanoFab for depositing exquisitely thin and highly uniform layers of metal—which Singh found crucial for making reliable electrical contacts—include both evaporation and sputtering, he says.

    The wide range of metals available for deposition at the NanoFab, uncommon at other nanotech facilities, was another draw.

    “We needed different metals compared to those commonly used on silicon wafers and the NanoFab provided those materials,” notes Singh.

    Topolancik, the Roche Sequencing Solutions engineer, needed high precision etching and deposition tools to fabricate a device that may ultimately improve cancer treatment. His company‘s plan to rapidly sequence DNA from cancer patients could quickly determine if potential anti-cancer drugs and those already in use are producing the genetic mutations necessary to fight cancer.

    “We want to know if the drug is working, and if not, to stop using it and change the treatment,” says Topolancik.

    In the standard method to sequence the double-stranded DNA molecule, a strand is peeled off and resynthesized, base by base, with each base—cytosine, adenine, guanine and thymine—tagged with a different fluorescent label.

    “It’s a very accurate but slow method,” says Topolancik.

    Instead of peeling apart the molecule, Topolancik is devising a method to read DNA directly, a much faster process. Borrowing a technique from the magnetic recording industry, he sandwiches the DNA between two electrodes separated by a gap just nanometers in width.


    Illustration of experiment to directly identify the base pairs of a DNA strand (denoted by A, C, T, G in graph). Tunneling current flows through DNA placed between two closely spaced electrodes. Different bases allow different amounts of current to flow, revealing the components of the DNA molecule.
    Credit: J. Topolancik/Roche Sequencing Solutions

    According to quantum theory, if the gap is small enough, electrons will spontaneously “tunnel” from one electrode to the other. In Topolancik’s setup, the tunneling electrons must pass through the DNA in order to reach the other electrode.

    The strength of the tunneling current identifies the bases of the DNA trapped between the electrodes. It’s an extremely rapid process, but for the technique to work reliably, the electrodes and the gap between them must be fabricated with extraordinarily high precision.

    That’s where the NanoFab comes in. To deposit layers of different metals just nanometers in thickness on a wafer, Topolancik relies on the NanoFab’s ion beam deposition tool. And to etch a pattern in those ultrathin, supersmooth layers without disturbing them—a final step in fabricating the electrodes—requires the NanoFab’s ion etching instrument.

    “These are specialty tools that are not usually accessible in academic facilities, but here [at the NanoFab] you have full, 24/7 access to them,” says Topolancik. “And if a tool goes down, it gets fixed right away,” he adds. “People here care about you, they want you to succeed because that’s the mission of the NanoFab.” As a result, he notes, “I can get done here in two weeks what would take half a year any place else.”

    Take a 360-degree walking tour of the CNST NanoFab in this video!

    See the full article here.

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    NIST Campus, Gaitherberg, MD, USA

    NIST Mission, Vision, Core Competencies, and Core Values

    NIST’s mission

    To promote U.S. innovation and industrial competitiveness by advancing measurement science, standards, and technology in ways that enhance economic security and improve our quality of life.
    NIST’s vision

    NIST will be the world’s leader in creating critical measurement solutions and promoting equitable standards. Our efforts stimulate innovation, foster industrial competitiveness, and improve the quality of life.
    NIST’s core competencies

    Measurement science
    Rigorous traceability
    Development and use of standards

    NIST’s core values

    NIST is an organization with strong values, reflected both in our history and our current work. NIST leadership and staff will uphold these values to ensure a high performing environment that is safe and respectful of all.

    Perseverance: We take the long view, planning the future with scientific knowledge and imagination to ensure continued impact and relevance for our stakeholders.
    Integrity: We are ethical, honest, independent, and provide an objective perspective.
    Inclusivity: We work collaboratively to harness the diversity of people and ideas, both inside and outside of NIST, to attain the best solutions to multidisciplinary challenges.
    Excellence: We apply rigor and critical thinking to achieve world-class results and continuous improvement in everything we do.

  • richardmitnick 7:46 am on May 17, 2017 Permalink | Reply
    Tags: Andrea Bertozzi, Mathematics, ,   

    From UCLA: Women in STEM-“UCLA innovator gets creative with applied mathematics” Andrea Bertozzi 

    UCLA bloc


    Andrea Bertozzi puts math to work solving real-world problems

    May 15, 2017
    Nico Correia

    UCLA mathematician Andrea Bertozzi works on a wide range of problems, ranging from the prediction of crime to the deployment of robotic bees. UCLA

    While her grade school classmates were learning the alphabet and how to count to five, Andrea Bertozzi remembers studying negative numbers and modular arithmetic.

    Math often gets a bad rap as an uncreative left brain-oriented activity, but Bertozzi recalls that, as a child, she was fascinated with it because of its creative potential.

    “Teachers have trouble teaching it that way,” said Bertozzi, a professor of mathematics and director of applied mathematics at UCLA, and the inaugural holder of UCLA’s Betsy Wood Knapp Chair for Innovation and Creativity. “They’re not looking at it the right way.”

    As the director of applied mathematics at UCLA and a member of the UCLA Institute for Digital Research and Education’s Executive Committee, Bertozzi and her colleagues conceive of math as a creative medium that can be practically used to solve real-world problems. “Our department is not one that does routine applications,” she said. “We develop new math on the boundary with other fields.”

    One of Bertozzi’s most publicized projects is an ideal illustration of math in action. In a partnership with the Los Angeles Police Department, Bertozzi and UCLA anthropology professor Jeffrey Brantingham head a research team that developed a mathematical model to predicts where and when crime will most likely happen, based on historical crime data in targeted areas so that police officers can preemptively patrol these districts.

    The model they and their team developed based on an algorithm that “learns,” evolves and adapts to new crime data is based on earthquake science. It takes a triggering event such as a property crime or a burglary and treats it similarly to aftershocks following an earthquake that can be tracked by scientists to figure out where and when the next one will occur.

    Another of Bertozzi’s projects, the deployment of robotic bees, is being done in collaboration with Spring Berman, a robotics expert and an assistant professor of mechanical and aerospace engineering at Arizona State University.

    This ground-based robotic bee was developed by undergraduates under Andrea Bertozzi’s direction to test algorithms needed to guide pollinating “bees” to designated plants.

    Since the late 1990s, the population of bees has plunged because of a combination of factors. Earlier this year, the rusty-patched bumblebee landed on the US Fish and Wildlife Service’s list of endangered species. Without bees to pollinate, humanity runs the risk of losing a wide swath of the world’s flora. One solution that scientists are looking into is the development of robotic bees.

    That’s where Bertozzi’s creative mathematical abilities come in.

    Bertozzi and Berman are studying algorithms that would send out a cloud of these robotic pollinators to certain plants. In the applied math lab at UCLA, undergraduates have created earthbound robotic bees to test path-planning algorithms for simple robots without GPS trackers. The group is planning to present the results of testbed simulation “flights” at a conference.

    Bertozzi isn’t exaggerating when she says she is working on a broad research agenda. Her interest in non-linear partial differential equations and applied mathematics has led to projects in everything from image-processing to cooperative robotics and high-dimensional data analysis.

    “It turns out that a lot of my recent projects have social components,” she said. “I have a lot of ideas; we work on those that I can pitch to the funding agencies.” She and her students have used a powerful computer resource at UCLA, the Hoffman2 Cluster, provided by the Institute of Digital Research and Education, to do their complex calculations.

    Although her research goals are all complex, Bertozzi has a concise philosophy on math.

    “You can think of math as a language that describes the real world,” said Bertozzi. “It’s about always reinventing and adding different structures to things.”

    See the full article here .

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    UC LA Campus

    For nearly 100 years, UCLA has been a pioneer, persevering through impossibility, turning the futile into the attainable.

    We doubt the critics, reject the status quo and see opportunity in dissatisfaction. Our campus, faculty and students are driven by optimism. It is not naïve; it is essential. And it has fueled every accomplishment, allowing us to redefine what’s possible, time after time.

    This can-do perspective has brought us 12 Nobel Prizes, 12 Rhodes Scholarships, more NCAA titles than any university and more Olympic medals than most nations. Our faculty and alumni helped create the Internet and pioneered reverse osmosis. And more than 100 companies have been created based on technology developed at UCLA.

  • richardmitnick 8:58 am on March 4, 2017 Permalink | Reply
    Tags: , , Making math more Lego-like, Mathematics   

    From Harvard: “Making math more Lego-like” 

    Harvard University
    Harvard University

    March 2, 2017
    Peter Reuell

    “[A] picture is worth 1,000 symbols,” quips Professor Arthur Jaffe (left). Jaffe and postdoctoral fellow Zhengwei Liu have developed a pictorial mathematical language that can convey pages of algebraic equations in a single 3-D drawing. Rose Lincoln/Harvard Staff Photographer

    Galileo called mathematics the “language with which God wrote the universe.” He described a picture-language, and now that language has a new dimension.

    The Harvard trio of Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski has developed a 3-D picture-language for mathematics with potential as a tool across a range of topics, from pure math to physics.

    Though not the first pictorial language of mathematics, the new one, called quon, holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images. The language is described in a February 2017 paper published in the Proceedings of the National Academy of Sciences.

    “It’s a big deal,” said Jacob Biamonte of the Quantum Complexity Science Initiative after reading the research. “The paper will set a new foundation for a vast topic.”

    “This paper is the result of work we’ve been doing for the past year and a half, and we regard this as the start of something new and exciting,” Jaffe said. “It seems to be the tip of an iceberg. We invented our language to solve a problem in quantum information, but we have already found that this language led us to the discovery of new mathematical results in other areas of mathematics. We expect that it will also have interesting applications in physics.”

    When it comes to the “language” of mathematics, humans start with the basics — by learning their numbers. As we get older, however, things become more complex.

    “We learn to use algebra, and we use letters to represent variables or other values that might be altered,” Liu said. “Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace ‘symbol proof’ by ‘picture proof.’”

    The new language relies on images to convey the same information that is found in traditional algebraic equations — and in some cases, even more.

    “An image can contain information that is very hard to describe algebraically,” Liu said. “It is very easy to transmit meaning through an image, and easy for people to understand what they see in an image, so we visualize these concepts and instead of words or letters can communicate via pictures.”

    “So this pictorial language for mathematics can give you insights and a way of thinking that you don’t see in the usual, algebraic way of approaching mathematics,” Jaffe said. “For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension.”

    In their most recent work, the researchers moved their language into a more literal realm, creating 3-D images that, when manipulated, can trigger mathematical insights.

    “Where before we had been working in two dimensions, we now see that it’s valuable to have a language that’s Lego-like, and in three dimensions,” Jaffe said. “By pushing these pictures around, or working with them like an object you can deform, the images can have different mathematical meanings, and in that way we can create equations.”

    Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.

    “It turns out one picture is worth 1,000 symbols,” Jaffe said.

    “We could describe this algebraically, and it might require an entire page of equations,” Liu added. “But we can do that in one picture, so it can capture a lot of information.”

    Having found a fit with quantum information, the researchers are now exploring how their language might also be useful in a number of other subjects in mathematics and physics.

    “We don’t want to make claims at this point,” Jaffe said, “but we believe and are thinking about quite a few other areas where this picture-language could be important.”

    See the full article here .

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    Harvard University campus

    Harvard is the oldest institution of higher education in the United States, established in 1636 by vote of the Great and General Court of the Massachusetts Bay Colony. It was named after the College’s first benefactor, the young minister John Harvard of Charlestown, who upon his death in 1638 left his library and half his estate to the institution. A statue of John Harvard stands today in front of University Hall in Harvard Yard, and is perhaps the University’s best known landmark.

    Harvard University has 12 degree-granting Schools in addition to the Radcliffe Institute for Advanced Study. The University has grown from nine students with a single master to an enrollment of more than 20,000 degree candidates including undergraduate, graduate, and professional students. There are more than 360,000 living alumni in the U.S. and over 190 other countries.

  • richardmitnick 3:27 pm on December 30, 2016 Permalink | Reply
    Tags: , , cohomology theories, Feynman diagrams, Mathematics, Motive - recurring theme, Periods of motion, , The fact that the periods that come from physics are “somehow God-given ...", Weight   

    From Quanta: “Strange Numbers Found in Particle Collisions” 

    Quanta Magazine
    Quanta Magazine

    November 15, 2016
    Kevin Hartnett

    Particle collisions are somehow linked to mathematical “motives.” Xiaolin Zeng for Quanta Magazine

    At the Large Hadron Collider in Geneva, physicists shoot protons around a 17-mile track and smash them together at nearly the speed of light.

    CERN/LHC Map
    CERN LHC Grand Tunnel
    CERN LHC particles
    LHC at CERN

    It’s one of the most finely tuned scientific experiments in the world, but when trying to make sense of the quantum debris, physicists begin with a strikingly simple tool called a Feynman diagram that’s not that different from how a child would depict the situation.

    James O’Brien for Quanta Magazine

    Feynman diagrams were devised by Richard Feynman in the 1940s. They feature lines representing elementary particles that converge at a vertex (which represents a collision) and then diverge from there to represent the pieces that emerge from the crash. Those lines either shoot off alone or converge again. The chain of collisions can be as long as a physicist dares to consider.

    To that schematic physicists then add numbers, for the mass, momentum and direction of the particles involved. Then they begin a laborious accounting procedure — integrate these, add that, square this. The final result is a single number, called a Feynman probability, which quantifies the chance that the particle collision will play out as sketched.

    “In some sense Feynman invented this diagram to encode complicated math as a bookkeeping device,” said Sergei Gukov, a theoretical physicist and mathematician at the California Institute of Technology.

    Feynman diagrams have served physics well over the years, but they have limitations. One is strictly procedural. Physicists are pursuing increasingly high-energy particle collisions that require greater precision of measurement — and as the precision goes up, so does the intricacy of the Feynman diagrams that need to be calculated to generate a prediction.

    The second limitation is of a more fundamental nature. Feynman diagrams are based on the assumption that the more potential collisions and sub-collisions physicists account for, the more accurate their numerical predictions will be. This process of calculation, known as perturbative expansion, works very well for particle collisions of electrons, where the weak and electromagnetic forces dominate. It works less well for high-energy collisions, like collisions between protons, where the strong nuclear force prevails. In these cases, accounting for a wider range of collisions — by drawing ever more elaborate Feynman diagrams — can actually lead physicists astray.

    “We know for a fact that at some point it begins to diverge” from real-world physics, said Francis Brown, a mathematician at the University of Oxford. “What’s not known is how to estimate at what point one should stop calculating diagrams.”

    Yet there is reason for optimism. Over the last decade physicists and mathematicians have been exploring a surprising correspondence that has the potential to breathe new life into the venerable Feynman diagram and generate far-reaching insights in both fields. It has to do with the strange fact that the values calculated from Feynman diagrams seem to exactly match some of the most important numbers that crop up in a branch of mathematics known as algebraic geometry. These values are called “periods of motives,” and there’s no obvious reason why the same numbers should appear in both settings. Indeed, it’s as strange as it would be if every time you measured a cup of rice, you observed that the number of grains was prime.

    “There is a connection from nature to algebraic geometry and periods, and with hindsight, it’s not a coincidence,” said Dirk Kreimer, a physicist at Humboldt University in Berlin.

    Now mathematicians and physicists are working together to unravel the coincidence. For mathematicians, physics has called to their attention a special class of numbers that they’d like to understand: Is there a hidden structure to these periods that occur in physics? What special properties might this class of numbers have? For physicists, the reward of that kind of mathematical understanding would be a new degree of foresight when it comes to anticipating how events will play out in the messy quantum world.

    Lucy Reading-Ikkanda for Quanta Magazine

    A Recurring Theme

    Today periods are one of the most abstract subjects of mathematics, but they started out as a more concrete concern. In the early 17th century scientists such as Galileo Galilei were interested in figuring out how to calculate the length of time a pendulum takes to complete a swing. They realized that the calculation boiled down to taking the integral — a kind of infinite sum — of a function that combined information about the pendulum’s length and angle of release. Around the same time, Johannes Kepler used similar calculations to establish the time that a planet takes to travel around the sun. They called these measurements “periods,” and established them as one of the most important measurements that can be made about motion.

    Over the course of the 18th and 19th centuries, mathematicians became interested in studying periods generally — not just as they related to pendulums or planets, but as a class of numbers generated by integrating polynomial functions like x^2 + 2x – 6 and 3x^3 – 4×2 – 2x + 6. For more than a century, luminaries like Carl Friedrich Gauss and Leonhard Euler explored the universe of periods and found that it contained many features that pointed to some underlying order. In a sense, the field of algebraic geometry — which studies the geometric forms of polynomial equations — developed in the 20th century as a means for pursuing that hidden structure.

    This effort advanced rapidly in the 1960s. By that time mathematicians had done what they often do: They translated relatively concrete objects like equations into more abstract ones, which they hoped would allow them to identify relationships that were not initially apparent.

    This process first involved looking at the geometric objects (known as algebraic varieties) defined by the solutions to classes of polynomial functions, rather than looking at the functions themselves. Next, mathematicians tried to understand the basic properties of those geometric objects. To do that they developed what are known as cohomology theories — ways of identifying structural aspects of the geometric objects that were the same regardless of the particular polynomial equation used to generate the objects.

    By the 1960s, cohomology theories had proliferated to the point of distraction — singular cohomology, de Rham cohomology, étale cohomology and so on. Everyone, it seemed, had a different view of the most important features of algebraic varieties.

    It was in this cluttered landscape that the pioneering mathematician Alexander Grothendieck, who died in 2014, realized that all cohomology theories were different versions of the same thing.

    “What Grothendieck observed is that, in the case of an algebraic variety, no matter how you compute these different cohomology theories, you always somehow find the same answer,” Brown said.

    That same answer — the unique thing at the center of all these cohomology theories — was what Grothendieck called a “motive.” “In music it means a recurring theme. For Grothendieck a motive was something which is coming again and again in different forms, but it’s really the same,” said Pierre Cartier, a mathematician at the Institute of Advanced Scientific Studies outside Paris and a former colleague of Grothendieck’s.

    Motives are in a sense the fundamental building blocks of polynomial equations, in the same way that prime factors are the elemental pieces of larger numbers. Motives also have their own data associated with them. Just as you can break matter into elements and specify characteristics of each element — its atomic number and atomic weight and so forth — mathematicians ascribe essential measurements to a motive. The most important of these measurements are the motive’s periods. And if the period of a motive arising in one system of polynomial equations is the same as the period of a motive arising in a different system, you know the motives are the same.

    “Once you know the periods, which are specific numbers, that’s almost the same as knowing the motive itself,” said Minhyong Kim, a mathematician at Oxford.

    One direct way to see how the same period can show up in unexpected contexts is with pi, “the most famous example of getting a period,” Cartier said. Pi shows up in many guises in geometry: in the integral of the function that defines the one-dimensional circle, in the integral of the function that defines the two-dimensional circle, and in the integral of the function that defines the sphere. That this same value would recur in such seemingly different-looking integrals was likely mysterious to ancient thinkers. “The modern explanation is that the sphere and the solid circle have the same motive and therefore have to have essentially the same period,” Brown wrote in an email.

    Feynman’s Arduous Path

    If curious minds long ago wanted to know why values like pi crop up in calculations on the circle and the sphere, today mathematicians and physicists would like to know why those values arise out of a different kind of geometric object: Feynman diagrams.

    Feynman diagrams have a basic geometric aspect to them, formed as they are from line segments, rays and vertices. To see how they’re constructed, and why they’re useful in physics, imagine a simple experimental setup in which an electron and a positron collide to produce a muon and an antimuon. To calculate the probability of that result taking place, a physicist would need to know the mass and momentum of each of the incoming particles and also something about the path the particles followed. In quantum mechanics, the path a particle takes can be thought of as the average of all the possible paths it might take. Computing that path becomes a matter of taking an integral, known as a Feynman path integral, over the set of all paths.

    Every route a particle collision could follow from beginning to end can be represented by a Feynman diagram, and each diagram has its own associated integral. (The diagram and its integral are one and the same.) To calculate the probability of a specific outcome from a specific set of starting conditions, you consider all possible diagrams that could describe what happens, take each integral, and add those integrals together. That number is the diagram’s amplitude. Physicists then square the magnitude of this number to get the probability.

    This procedure is easy to execute for an electron and a positron going in and a muon and an antimuon coming out. But that’s boring physics. The experiments that physicists really care about involve Feynman diagrams with loops. Loops represent situations in which particles emit and then reabsorb additional particles. When an electron collides with a positron, there’s an infinite number of intermediate collisions that can take place before the final muon and antimuon pair emerges. In these intermediate collisions, new particles like photons are created and annihilated before they can be observed. The entering and exiting particles are the same as previously described, but the fact that those unobservable collisions happen can still have subtle effects on the outcome.

    “It’s like Tinkertoys. Once you draw a diagram you can connect more lines according to the rules of the theory,” said Flip Tanedo, a physicist at the University of California, Riverside. “You can connect more sticks, more nodes, to make it more complicated.”

    By considering loops, physicists increase the precision of their experiments. (Adding a loop is like calculating a value to a greater number of significant digits). But each time they add a loop, the number of Feynman diagrams that need to be considered — and the difficulty of the corresponding integrals — goes up dramatically. For example, a one-loop version of a simple system might require just one diagram. A two-loop version of the same system needs seven diagrams. Three loops demand 72 diagrams. Increase it to five loops, and the calculation requires around 12,000 integrals — a computational load that can literally take years to resolve.

    Rather than chugging through so many tedious integrals, physicists would love to gain a sense of the final amplitude just by looking at the structure of a given Feynman diagram — just as mathematicians can associate periods with motives.

    “This procedure is so complex and the integrals are so hard, so what we’d like to do is gain insight about the final answer, the final integral or period, just by staring at the graph,” Brown said.

    Lucy Reading-Ikkanda for Quanta Magazine

    A Surprising Connection

    Periods and amplitudes were presented together for the first time in 1994 by Kreimer and David Broadhurst, a physicist at the Open University in England, with a paper following in 1995. The work led mathematicians to speculate that all amplitudes were periods of mixed Tate motives — a special kind of motive named after John Tate, emeritus professor at Harvard University, in which all the periods are multiple values of one of the most influential constructions in number theory, the Riemann zeta function. In the situation with an electron-positron pair going in and a muon-antimuon pair coming out, the main part of the amplitude comes out as six times the Riemann zeta function evaluated at three.

    If all amplitudes were multiple zeta values, it would give physicists a well-defined class of numbers to work with. But in 2012 Brown and his collaborator Oliver Schnetz proved that’s not the case. While all the amplitudes physicists come across today may be periods of mixed Tate motives, “there are monsters lurking out there that throw a spanner into the works,” Brown said. Those monsters are “certainly periods, but they’re not the nice and simple periods people had hoped for.”

    What physicists and mathematicians do know is that there seems to be a connection between the number of loops in a Feynman diagram and a notion in mathematics called “weight.” Weight is a number related to the dimension of the space being integrated over: A period integral over a one-dimensional space can have a weight of 0, 1 or 2; a period integral over a two-dimensional space can have weight up to 4, and so on. Weight can also be used to sort periods into different types: All periods of weight 0 are conjectured to be algebraic numbers, which can be the solutions to polynomial equations (this has not been proved); the period of a pendulum always has a weight of 1; pi is a period of weight 2; and the weights of values of the Riemann zeta function are always twice the input (so the zeta function evaluated at 3 has a weight of 6).

    This classification of periods by weights carries over to Feynman diagrams, where the number of loops in a diagram is somehow related to the weight of its amplitude. Diagrams with no loops have amplitudes of weight 0; the amplitudes of diagrams with one loop are all periods of mixed Tate motives and have, at most, a weight of 4. For graphs with additional loops, mathematicians suspect the relationship continues, even if they can’t see it yet.

    “We go to higher loops and we see periods of a more general type,” Kreimer said. “There mathematicians get really interested because they don’t understand much about motives that are not mixed Tate motives.”

    Mathematicians and physicists are currently going back and forth trying to establish the scope of the problem and craft solutions. Mathematicians suggest functions (and their integrals) to physicists that can be used to describe Feynman diagrams. Physicists produce configurations of particle collisions that outstrip the functions mathematicians have to offer. “It’s quite amazing to see how fast they’ve assimilated quite technical mathematical ideas,” Brown said. “We’ve run out of classical numbers and functions to give to physicists.”

    Nature’s Groups

    Since the development of calculus in the 17th century, numbers arising in the physical world have informed mathematical progress. Such is the case today. The fact that the periods that come from physics are “somehow God-given and come from physical theories means they have a lot of structure and it’s structure a mathematician wouldn’t necessarily think of or try to invent,” said Brown.

    Adds Kreimer, “It seems so that the periods which nature wants are a smaller set than the periods mathematics can define, but we cannot define very cleanly what this subset really is.”

    Brown is looking to prove that there’s a kind of mathematical group — a Galois group — acting on the set of periods that come from Feynman diagrams. “The answer seems to be yes in every single case that’s ever been computed,” he said, but proof that the relationship holds categorically is still in the distance. “If it were true that there were a group acting on the numbers coming from physics, that means you’re finding a huge class of symmetries,” Brown said. “If that’s true, then the next step is to ask why there’s this big symmetry group and what possible physics meaning could it have.”

    Among other things, it would deepen the already provocative relationship between fundamental geometric constructions from two very different contexts: motives, the objects that mathematicians devised 50 years ago to understand the solutions to polynomial equations, and Feynman diagrams, the schematic representation of how particle collisions play out. Every Feynman diagram has a motive attached to it, but what exactly the structure of a motive is saying about the structure of its related diagram remains anyone’s guess.

    See the full article here .

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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.

  • richardmitnick 12:32 pm on September 12, 2016 Permalink | Reply
    Tags: , , , , Mathematics,   

    From CSIRO: Women in STEM – “Indigenous STEM Awards: discovering our stars” Karlie Noon 

    CSIRO bloc

    Commonwealth Scientific and Industrial Research Organisation

    Karlie’s love of mathematics and astronomy has made her a star in her field.

    You could say ‘it was written in the stars’ for Karlie Noon to come and work for us, but like all good stories it can’t be summed up in a simple sentence.

    Karlie is helping us look for an outstanding Indigenous science, technology, engineering or maths professional to inspire the next generation of young innovators.

    She has a tattoo of the solar system – the sun on her left shoulder, all nine planets, stars and an asteroid belt heading down to her wrist. Not only does it symbolise many hours in the tattooist’s chair, for its wearer, it’s a symbol of her journey.

    Karlie is a 26-year-old Kamilaroi woman who was born and raised in Tamworth. She grew up speaking Kamilaroi and Waradjuri and, by her own admissions, spent more time at home playing games than studying in the classroom.

    While schooling wasn’t an important part of her childhood, it was clear that mathematics was. From being taught once a week at home by a local elder to studying pure maths (the why of maths) at the University of Newcastle, her love of mathematics has been a constant.

    “When I was growing up I didn’t really think that English was important, which is funny because a lot of kids can’t see why maths is important,” Karlie said.

    “But for me, maths was always the most interesting and most important thing to study – and I still love it.”

    It was perhaps a gravitational pull of science and maths that made her change her study path from philosophy to astronomy. As the first in her family to attend university the tattoo was a gift to herself – and it was one of the first things esteemed astronomer Dr Duane Hamacher noticed about Karlie.

    You could say the planets aligned when Dr Hamacher and Karlie were in the same place at the same time, and while planets do align occasionally, any astronomer would know that saying is more fanciful than scientific. Dr Hamacher got the sense that Karlie had more than a little interest in astronomy if she was dedicated enough to get such a tattoo.

    Karlie wants to inspire the next generation of young innovators.

    They met through the BHP Billiton Foundation-funded Aboriginal Summer School for Excellence in Technology and Science (ASSETS). Karlie was a cultural mentor at the ASSETS program, an experience she absolutely loved. She landed the role after applying for an Indigenous Cadetship with us, and would not have missed the experience for the world.

    Eight months after ASSETS 2016 came to a close, Karlie is now researching Indigenous Astronomy with Dr Hamacher. The two share an enthusiasm for the cosmological and Karlie has some amazing insights to offer on the Indigenous perspectives of the study.

    We noticed Karlie’s dedication and skills too. She is now working for us to establish the Indigenous STEM Awards program designed to recognise, reward and celebrate Aboriginal and Torres Strait Islander students and scientists. The awards program has seven individual awards divided into three categories rewarding students, teachers, schools, communities and STEM professionals. For Karlie, she hopes the awards can inspire Aboriginal and Torres Strait Islander students across the country.

    “I want to get the word out that Aboriginal and Torres Strait Islander students are natural scientists – and that doesn’t mean we are good at studying nature, it means we are good at investigating,” Karlie said.

    “There is so much we can do and we are so unique in our approach and that is incredibly valuable in science and STEM.”

    “I’m really looking forward to seeing some amazing students, teachers, schools and professionals applying for the awards, and can’t wait to read their stories.”

    See the full article here .

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    CSIRO campus

    CSIRO, the Commonwealth Scientific and Industrial Research Organisation, is Australia’s national science agency and one of the largest and most diverse research agencies in the world.

  • richardmitnick 9:34 am on August 29, 2016 Permalink | Reply
    Tags: , , Mapping signal paths in proteins could reveal new direction for drug development, Mathematics   

    From ICL: “Mapping signal paths in proteins could reveal new direction for drug development” 

    Imperial College London
    Imperial College London

    26 August 2016
    Hayley Dunning

    Using math previously applied to traffic jams and electrical grids, researchers have developed a new method to map signal propagation in proteins. No image credit

    Proteins are molecules found within every cell in the human body that carry out a wide range of functions essential for life. Many proteins have an ‘active site’ to which other molecules bind, enabling them to perform different functions, such as catalysing biochemical reactions or regulating gene expression.

    Active sites are often targeted by drugs designed to combat a host of diseases caused by malfunctioning proteins.

    However, some proteins also have additional sites to which other molecules bind, causing the protein to shift its shape and altering the functionality of the main active site.

    For example, a protein can be ‘activated’ or ‘deactivated’ through this additional binding. This process is known as ‘allostery’, and these additional allosteric sites are often far away from the main active site in the structure of the protein.

    Many proteins are known to have allosteric sites, and these are crucial to biological function. However, the big mystery has been how to predict if and where such allosteric sites exist, and how signals travel across the protein from allosteric sites to the active site.

    Now, researchers at Imperial College London have used sophisticated mathematical methods to accurately trace the allosteric signals through proteins. Their method, published today in Nature Communications, not only allows them to track the signal by identifying the chemical bonds involved, but also predict new allosteric sites.

    New drug targets

    Allosteric sites are a potentially exciting new target for drugs, since they allow greater flexibility than active sites. The structure of active sites may be shared across several proteins, meaning any drugs targeting that particular structure could have side effects, whereas allosteric sites are more specialised and targetting them could minimise unwanted interferences.

    Study co-author, Professor Mauricio Barahona from the Department of Mathematics at Imperial has been working on the underlying mathematical tools, and has already applied them to the study of traffic jams and cascading failures in electrical grids.

    He said: “The concept is the same in all these cases: we look at how a signal travels within the graph structure, whether that’s cars in the road network of a city, electricity in the power grid, or fluctuations in the chemicals bonds in the structure of a protein.

    “When a line is tripped in a power grid, it can have its largest effect on a distant part of the network. The same principle is at play in allostery.”

    Professor Sophia Yaliraki from the Department of Chemistry at Imperial, who has been working on the underlying chemical theory, added: “The purpose of modelling in each case is to figure out how to interfere with the signal – either to enhance it or disrupt it. Disrupting the signal in proteins could inhibit their function, effectively targeting diseases where proteins are malfunctioning.

    “This depends both on the specific atomic-scale structure of the protein, as well as its overall three-dimensional shape.”

    Mapping influencers

    The mathematical models work by mapping influencers – in this case which chemical bonds influence other bonds in response to a propagating signal from the active site. Despite the large amount of information required, the computational method is “incredibly efficient” according to Professor Yaliraki, allowing signal pathways in large complex proteins to be mapped in minutes.

    The researchers have applied the model to many known allosteric sites, and found they were able to accurately predict their existence and position. Now, they are applying the methodology to proteins that are not yet known for allostery in the hope of identifying new targets for drug development.

    The work is a collaboration between researchers in the Departments of Chemistry and Mathematics, enabled by the EPSRC-funded cross-disciplinary Institute for Chemical Biology at Imperial.

    See the full article here .

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    Imperial College London

    Imperial College London is a science-based university with an international reputation for excellence in teaching and research. Consistently rated amongst the world’s best universities, Imperial is committed to developing the next generation of researchers, scientists and academics through collaboration across disciplines. Located in the heart of London, Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.

  • richardmitnick 11:21 am on June 10, 2016 Permalink | Reply
    Tags: , , , Laura Shou, Mathematics,   

    From Caltech: Women in Science “Shou Receives Fellowship for Graduate Studies in Germany” Laura Shou 

    Caltech Logo

    Lori Dajose

    Laura Shou. Credit: Courtesy of L. Shou

    Laura Shou, a senior in mathematics, has received a Graduate Study Scholarship from the German Academic Exchange Service (DAAD) to pursue a master’s degree in Germany. She will spend one year at the Ludwig-Maximilians-Universität München and the Technische Universität München, studying in the theoretical and mathematical physics (TMP) program.

    The DAAD is the German national agency for the support of international academic cooperation. The organization aims to promote international academic relations and cooperation by offering mobility programs for students, faculty, and administrators and others in the higher education realm. The Graduate Study Scholarship supports highly qualified American and Canadian students with an opportunity to conduct independent research or complete a full master’s degree in Germany. Master’s scholarships are granted for 12 months and are eligible for up to a one-year extension in the case of two-year master’s programs. Recipients receive a living stipend, health insurance, educational costs, and travel.

    “As a math major, I was especially interested in the TMP course because of its focus on the interplay between theoretical physics and mathematics,” Shou says. “I would like to use mathematical rigor and analysis to work on problems motivated by physics. The TMP course at the LMU/TUM is one of the few programs focused specifically on mathematical physics. There are many people doing research in mathematical physics there, and the program also regularly offers mathematically rigorous physics classes.”

    At Caltech, Shou has participated in the Summer Undergraduate Research Fellowship (SURF) program three times, conducting research with Professor of Mathematics Yi Ni on knot theory and topology, with former postdoctoral fellow Chris Marx (PhD ’12) on mathematical physics, and with Professor of Mathematics Nets Katz on analysis. She was the president of the Dance Dance Revolution Club and a member of the Caltech NERF Club and the Caltech Math Club.

    Following her year in Germany, Shou will begin the mathematics PhD program at Princeton.

    See the full article here .

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    Caltech campus

    The California Institute of Technology (commonly referred to as Caltech) is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering. Its 124-acre (50 ha) primary campus is located approximately 11 mi (18 km) northeast of downtown Los Angeles. “The mission of the California Institute of Technology is to expand human knowledge and benefit society through research integrated with education. We investigate the most challenging, fundamental problems in science and technology in a singularly collegial, interdisciplinary atmosphere, while educating outstanding students to become creative members of society.”

  • richardmitnick 11:57 am on May 30, 2016 Permalink | Reply
    Tags: , Mathematics, , Two-hundred-terabyte maths proof is largest ever   

    From Nature: “Two-hundred-terabyte maths proof is largest ever” 

    Nature Mag

    26 May 2016
    Evelyn Lamb

    U Texas Stampede Supercomputer. Texas Advanced Computer Center
    U Texas Stampede Supercomputer. Texas Advanced Computer Center

    Three computer scientists have announced the largest-ever mathematics proof: a file that comes in at a whopping 200 terabytes1, roughly equivalent to all the digitized text held by the US Library of Congress. The researchers have created a 68-gigabyte compressed version of their solution — which would allow anyone with about 30,000 hours of spare processor time to download, reconstruct and verify it — but a human could never hope to read through it.

    Computer-assisted proofs too large to be directly verifiable by humans have become commonplace, and mathematicians are familiar with computers that solve problems in combinatorics — the study of finite discrete structures — by checking through umpteen individual cases. Still, “200 terabytes is unbelievable”, says Ronald Graham, a mathematician at the University of California, San Diego. The previous record-holder is thought to be a 13-gigabyte proof2, published in 2014.

    The puzzle that required the 200-terabyte proof, called the Boolean Pythagorean triples problem, has eluded mathematicians for decades. In the 1980s, Graham offered a prize of US$100 for anyone who could solve it. (He duly presented the cheque to one of the three computer scientists, Marijn Heule of the University of Texas at Austin, earlier this month.) The problem asks whether it is possible to colour each positive integer either red or blue, so that no trio of integers a, b and c that satisfy Pythagoras’ famous equation a2 + b2 = c2 are all the same colour. For example, for the Pythagorean triple 3, 4 and 5, if 3 and 5 were coloured blue, 4 would have to be red.

    In a paper* posted on the arXiv server on 3 May, Heule, Oliver Kullmann of Swansea University, UK, and Victor Marek of the University of Kentucky in Lexington have now shown that there are many allowable ways to colour the integers up to 7,824 — but when you reach 7,825, it is impossible for every Pythagorean triple to be multicoloured1. There are more than 102,300 ways to colour the integers up to 7,825, but the researchers took advantage of symmetries and several techniques from number theory to reduce the total number of possibilities that the computer had to check to just under 1 trillion. It took the team about 2 days running 800 processors in parallel on the University of Texas’s Stampede supercomputer to zip through all the possibilities. The researchers then verified the proof using another computer program.

    The numbers 1 to 7,824 can be coloured either red or blue so that no trio a, b and c that satisfies a2 +b2 = c2 is all the same colour. The grid of 7,824 squares here shows one such solution, with numbers coloured red or blue (a white square can be either). But for the numbers 1 to 7,825, there is no solution.

    Facts vs theory

    The Pythagorean triples problem is one of many similar questions in Ramsey theory, an area of mathematics that is concerned with finding structures that must appear in sufficiently large sets. For example, the researchers think that if the problem had allowed three colours, rather than two, they would still hit a point where it would be impossible to avoid creating a Pythagorean triple where a, b and c were all the same colour; indeed, they conjecture that this is the case for any finite choice of colours. Any proof for more colours will probably be much larger even than the 200-terabyte 2-colour proof, unless researchers can simplify the case-by-case checking process with a breakthrough in understanding.

    Although the computer solution has cracked the Boolean Pythagorean triples problem, it hasn’t provided an underlying reason why the colouring is impossible, or explored whether the number 7,825 is meaningful, says Kullmann. That echoes a common philosophical objection to the value of computer-assisted proofs: they may be correct, but are they really mathematics? If mathematicians’ work is understood to be a quest to increase human understanding of mathematics, rather than to accumulate an ever-larger collection of facts, a solution that rests on theory seems superior to a computer ticking off possibilities.

    hat did ultimately occur in the case of the 13-gigabyte proof from 2014, which solved a special case of a question called the Erdős discrepancy problem. A year later, mathematician Terence Tao of the University of California, Los Angeles, solved the general problem the old-fashioned way3 — a much more satisfying resolution.

    Nature doi:10.1038/nature.2016.19990

    Science paper:
    Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer


    Heule, M. J. H., Kullmann, O. & Marek, V. W. Preprint at http://arxiv.org/abs/1605.00723 (2016).
    Show context

    Konev, B. & Lisitsa, A. Preprint at http://arxiv.org/abs/1402.2184 (2014).
    Show context

    Tao, T. Preprint at http://arxiv.org/abs/1509.05363 (2015).
    Show context

    For references, see the original article

    See the full article here .

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    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 4:42 pm on July 28, 2015 Permalink | Reply
    Tags: , , Mathematics,   

    From NYT: “The Singular Mind of Terry Tao’ 

    New York Times

    The New York Times

    JULY 24, 2015


    A prodigy grows up to become one of the greatest mathematicians in the world.

    This April, as undergraduates strolled along the street outside his modest office on the campus of the University of California, Los Angeles, the mathematician Terence Tao mused about the possibility that water could spontaneously explode. A widely used set of equations describes the behavior of fluids like water, but there seems to be nothing in those equations, he told me, that prevents a wayward eddy from suddenly turning in on itself, tightening into an angry gyre, until the density of the energy at its core becomes infinite: a catastrophic ‘‘singularity.’’ Someone tossing a penny into the fountain by the faculty center or skipping a stone at the Santa Monica beach could apparently set off a chain reaction that would take out Southern California.

    This doesn’t tend to happen. And yet, Tao explained, nobody can say precisely why. It’s a decades-­old conundrum, and Tao has recently been working on an approach to a solution — one part fanciful, one part outright absurd, like some lost passage from ‘‘Alice’s Adventures in Wonderland.’’

    Imagine, he said, that someone awfully clever could construct a machine out of pure water. It would be built not of rods and gears but from a pattern of interacting currents. As he talked, Tao carved shapes in the air with his hands, like a magician. Now imagine, he went on, that this machine were able to make a smaller, faster copy of itself, which could then make another, and so on, until one ‘‘has infinite speed in a tiny space and blows up.’’ Tao was not proposing constructing such a machine — ‘‘I don’t know how!’’ he said, laughing. It was merely a thought experiment, of the sort that [Albert] Einstein used to develop the theory of special relativity. But, Tao explained, if he can show mathematically that there is nothing, in principle, preventing such a fiendish contraption from operating, then it would mean that water can, in fact, explode. And in the process, he will have also solved the Navier-­Stokes global regularity problem, which has become, since it emerged more than a century ago, one of the most important in all of mathematics.

    Tao, who is 40, sat at a desk by the window, papers lying in drifts at the margins. Thin and unassuming, he was dressed in Birkenstocks, a rumpled blue-gray polo shirt and jeans with the cuffs turned up. Behind him, a small almond couch faced a glyph-­covered blackboard running the length of the room. The couch had been pulled away from the wall to accommodate the beat-up Trek bike he rides to work. At the room’s other end stood a fiberboard bookcase haphazardly piled with books, including ‘‘Compactness and Contradiction’’ and ‘‘Poincaré’s Legacies, Part I,’’ two of the 16 volumes Tao has written since he was a teenager.

    Fame came early for Tao, who was born in South Australia. An old headline in his hometown paper, The Advertiser, reads: ‘‘TINY TERENCE, 7, IS HIGH-SCHOOL WHIZ.’’ The clipping includes a photo of a diminutive Tao in 11th-grade math class, wearing a V-neck sweater over a white turtleneck, kneeling on his chair so he can reach a desk he is sharing with a girl more than twice his age. His teacher told the reporter that he hardly taught Tao anything, because Tao was always working two lessons ahead of the others. (Tao taught himself to read at age 2.)

    A few months later, halfway through the school year, Tao was moved up to 12th-grade math. Three years later, at age 10, Tao became the youngest person in history to win a medal in the International Mathematical Olympiad. He has since won many other prizes, including a MacArthur ‘‘genius’’ grant and the Fields Medal, considered the Nobel Prize for mathematicians. Today, many regard Tao as the finest mathematician of his generation.

    That spring day in his office, reflecting on his career so far, Tao told me that his view of mathematics has utterly changed since childhood. ‘‘When I was growing up, I knew I wanted to be a mathematician, but I had no idea what that entailed,’’ he said in a lilting Australian accent. ‘‘I sort of imagined a committee would hand me problems to solve or something.’’ But it turned out that the work of real mathematicians bears little resemblance to the manipulations and memorization of the math student. Even those who experience great success through their college years may turn out not to have what it takes. The ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians. Tao now believes that his younger self, the prodigy who wowed the math world, wasn’t truly doing math at all. ‘‘It’s as if your only experience with music were practicing scales or learning music theory,’’ he said, looking into light pouring from his window. ‘‘I didn’t learn the deeper meaning of the subject until much later.’’

    Possibly the greatest mathematician since antiquity was Carl Friedrich Gauss, a dour German born in the late 18th century. He did not get along with his own children and kept important results to himself, seeing them as unsuitable for public view. They were discovered among his papers after his death. Before and since, the annals of the field have teemed with variations on this misfit theme, from Isaac Newton, the loner with a savage temper; to John Nash, the ‘‘beautiful mind’’ whose work shaped economics and even political science, but who was racked by paranoid delusions; to, more recently, ­Grigory Perelman, the Russian who conquered the Poincaré conjecture alone, then refused the Fields Medal, and who also allowed his fingernails to grow until they curled.

    Tao, by contrast, is, as one colleague put it, ‘‘super-normal.’’ He has a gentle, self-­deprecating manner. He eschews job offers from prestigious East Coast institutions in favor of a relaxed, no-drama department in a place where he can enjoy the weather. In class, he conveys a sense that mathematics is fun. One of his students told me that he had recently joked with another about the many ways Tao defies all the Hollywood mad-­genius tropes. ‘‘They will never make a movie about him,’’ he said. ‘‘He doesn’t have a troubled life. He has a family, and they seem happy, and he’s usually smiling.’’

    This can be traced to his own childhood, which he experienced as super-normal, even if, to outside eyes, it was anything but. Tao’s family spent most of his early years living in the foothills south of Adelaide, in a brick split-­level with views of Gulf St. Vincent. The home was designed by his father, Billy, a pediatrician who immigrated with Tao’s mother, Grace, from Hong Kong in 1972, three years before Tao, the eldest of three, was born in 1975. The three boys — Nigel, Trevor and ‘‘Terry,’’ as everyone calls him — often played together, and a favorite pastime was inventing board games. They typically appropriated a Scrabble board for a basic grid, then brought in Scrabble tiles, chess pieces, Chinese checkers, mah-jongg tiles and Dungeons & Dragons dice, according to Nigel, who now works for Google. For story lines, they frequently drew from video games coming out at the time, like Super Mario Bros., then added layers of complex, whimsical rules. (Trevor, a junior chess champion, was too good to beat, so the boys created a variation on that game as well: Each turn began with a die roll to determine which piece could be moved.) Tao was a voracious consumer of fantasy books like Terry Pratchett’s Discworld series. When a class was boring, he doodled intricate maps of imaginary lands.

    Terry Tao, age 7, in an 11th-grade math class. Credit Photograph by The Advertiser, from the Tao family

    By the spring of 1985, with a 9-year-old Tao splitting time between high school and nearby Flinders University, Billy and Grace took him on a three-week American tour to seek advice from top mathematicians and education experts. On the Baltimore campus of Johns Hopkins, they met with Julian Stanley, a Georgia-­born psychologist who founded the Center for Talented Youth there. Tao was one of the most talented math students Stanley ever tested — at 8 years old, Tao scored a 760 on the math portion of the SAT — but Stanley urged the couple to keep taking things slow and give their son’s emotional and social skills time to develop.

    Even at a relatively deliberate pace, by age 17, Tao had finished a master’s thesis (‘‘Convolution Operators Generated by Right-­Monogenic and Harmonic Kernels’’) and moved to Princeton University to start on his Ph.D. Tao’s application to the university included a letter from Paul Erdos, the revered Hungarian mathematician. ‘‘I am sure he will develop into a first-rate mathematician and perhaps into a really great one,’’ read Erdos’s brief, typewritten note. ‘‘I recommend him in the highest possible terms.’’ Yet on arrival, it was Tao, the teenage prodigy, who was intimidated. During Tao’s first year, Andrew Wiles, then a Princeton professor, announced that he proved Fermat’s Last Theorem, a legendary problem that had gone unsolved for more than three centuries. Tao’s fellow graduate students spoke eloquently about mathematical fields of which he had barely heard.

    Tao became notorious for his nights haunting the graduate computer room to play the historical-­simulation game Civilization. (He now avoids computer games, he told me, because of what he calls a ‘‘completist streak’’ that makes it hard to stop playing.) At a local comic-book store, Tao met a circle of friends who played ‘‘Magic: The Gathering,’’ the intricate fantasy card game. This was Tao’s first real experience hanging out with people his age, but there was also an element, he admitted, of escaping the pressures of Princeton. Gifted children often avoid challenges at which they might not excel. Before Tao went to Princeton, his grades had flagged at Flinders. In a course on quantum physics, the instructor told the class that the final would include an essay on the history of the field. Tao, then 12, blew off studying, and when he sat down for the exam, he was stunned to discover that the essay would count for half the grade. ‘‘I remember crying,’’ Tao said, ‘‘and the proctor had to escort me out.’’ He failed.

    At Princeton, crisis came in the form of the ‘‘generals,’’ a wide-­ranging, arduous oral examination administered by three professors. While other students spent months working through problem sets and giving one another mock exams, Tao settled on his usual test-prep strategy: last-­minute cramming. ‘‘I went in and very quickly got out of my depth,’’ he said. ‘‘They were asking questions which I had no ability to answer.’’ Immediately after, Tao sat with his adviser, Elias Stein, and felt that he had let him down. Tao wasn’t really trying, and the hardest part was yet to come.

    The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof. It is common to fill page after page with an attempt, the seasons turning, only to arrive precisely where you began, empty-handed — or to realize that a subtle flaw of logic doomed the whole enterprise from its outset. The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’ The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, ‘‘he crushes you.’’ So you take back moves and try something different, and he crushes you again, ‘‘in much the same way.’’ If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

    As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.

    Within his field, Tao is best known for a proof about a remarkable set of numbers known as the primes. The primes are the whole numbers larger than 1 that can be divided evenly by only themselves and 1. Thus, the first few primes are 2, 3, 5, 7 and 11. The number 4 is not a prime because it divides evenly by 2; the number 9 fails because it can be divided by 3. Prime numbers are fundamental building blocks in mathematics. Like the chemical elements, they combine to form a universe. To a chemist, water is two atoms of hydrogen and one of oxygen. Similarly, in mathematics, the number 12 is composed of two ‘‘atoms’’ of 2 and one ‘‘atom’’ of 3 (12 = 2 x 2 x 3).

    The primes are elementary and, at the same time, mysterious. They are a result of simple logic, yet they seem to appear at random on the number line; you never know when the next one will occur. They are at once orderly and disorderly. They have been incorporated into mysticism and religious ritual and have inspired works of music and even an Italian novel, ‘‘The Solitude of Prime Numbers.’’ It is easy to see why mathematicians consider the primes to be one of the universe’s foundations. From counting, you can develop the concept of number, and then, quite naturally, the basic operations of arithmetic: addition, subtraction, multiplication and division. That is all you need to spot the primes — though, eerily, scientists have uncovered deep connections between primes and quantum mechanics that remain unexplained. Imagine that there is an advanced civilization of aliens around some distant star: They surely do not speak English, they may or may not have developed television, but we can be almost certain that their mathematicians have discovered the primes and puzzled over them.

    Tao’s work is related to the twin-prime conjecture, which the French mathematician Alphonse de Polignac suggested in 1849. Go up the number line, circling the primes, and you may notice that sometimes a pair of primes is separated by just 2: 5 and 7, 11 and 13, 17 and 19. These are the ‘‘twin primes,’’ and as the journey along the number line continues, they occur less frequently: 2,237 and 2,239 are followed by 2,267 and 2,269; after 31,391 and 31,393, there isn’t another pair until you reach 31,511 and 31,513. Euclid devised a simple, beautiful proof showing that there is an infinite number of primes. But what of the twin primes? As far as you go on the number line, will there always be another set of twins? The conjecture has roundly defeated all attempts at proving it.

    When mathematicians face a question they cannot answer, they sometimes devise a less stringent question, in the hope that solving it will provide insights. This is the path that Tao took in 2004, in collaboration with Ben Green of Oxford. Twins are two primes that are separated by exactly 2, but Green and Tao considered a looser definition, strings of primes separated by a constant, be it 2 or any other number. (For example, the primes 3, 7 and 11 are separated by the constant 4.) They sought to prove that no matter how long a string someone found, there would always be another longer string with a constant gap between its primes. That February, after some initial conversations, Green came to visit Tao at U.C.L.A., and in just two exhilarating months, they completed what is now known as the Green-Tao theorem. It could be a way point on the path to the twin-prime conjecture, and it forged deep connections between disparate areas of math, helping establish an interdisciplinary area called additive combinatorics. ‘‘It opened a lot of new directions in research,’’ says Izabella Laba, a University of British Columbia mathematician who has worked with Tao. ‘‘It gave a lot of people a lot of things to do.’’

    This sort of collaboration has been a hallmark of Tao’s career. Most mathematicians tend to specialize, but Tao ranges widely, learning from others and then working with them to make discoveries. Markus Keel, a longtime collaborator and close friend, reaches to science fiction to explain Tao’s ability to rapidly digest and employ mathematical ideas: Seeing Tao in action, Keel told me, reminds him of the scene in ‘‘The Matrix’’ when Neo has martial arts downloaded into his brain and then, opening his eyes, declares, ‘‘I know kung fu.’’ The citation for Tao’s Fields Medal, awarded in 2006, is a litany of boundary hopping and notes particularly ‘‘beautiful work’’ on Horn’s conjecture, which Tao completed with a friend he had played foosball with in graduate school. It was a new area of mathematics for Tao, at a great remove from his known stamping grounds. ‘‘This is akin,’’ the citation read, ‘‘to a leading English-­language novelist suddenly producing the definitive Russian novel.’’

    The Green-Tao theorem on primes was a similar collaboration. Green is a specialist in an area called number theory, and Tao originally trained in an area called harmonic analysis. Yet, as they told me, the proof depended on the insights of many other mathematicians. In the game of devil’s chess, players have no real hope if they haven’t studied the winning games of the masters. A proof establishes facts that can be used in subsequent proofs, but it also offers a set of moves and strategies that forced the devil to submit — a devious way to pin one of his pieces or shut down a counterattack, or an endgame move that sacrifices a bishop to gain a winning position. Just as a chess player might examine variations of the Ruy Lopez and King’s Indian Defense, a mathematician might study particularly clever applications of the Chinese remainder theorem or the sieve of Eratosthenes. The wise player has a vast repertoire to draw on, and the crafty player intuits the move that suits the moment.

    For their work, Tao and Green salvaged a crucial bit from an earlier proof done by others, which had been discarded as incorrect, and aimed at a different goal. Other maneuvers came from masterful proofs by Timothy Gowers of England and Endre Szemeredi of Hungary. Their work, in turn, relied on contributions from Erdos, Klaus Roth and Frank Ramsey, an Englishman who died at age 26 in 1930, and on and on, into history. Ask mathematicians about their experience of the craft, and most will talk about an intense feeling of intellectual camaraderie. ‘‘A very central part of any mathematician’s life is this sense of connection to other minds, alive today and going back to Pythagoras,’’ said Steven Strogatz, a professor of mathematics at Cornell University. ‘‘We are having this conversation with each other going over the millennia.’’

    The Green-Tao theorem caught the mathematical community by surprise, because that problem was thought to be many years from succumbing to proof. On the day I visited Tao, we ate lunch on the outdoor patio of the midcentury-­modern faculty center. Working on a modest plate of sushi, Tao told me that he and Green have continued to work around the margins of the twin-prime conjecture, as have others, with a lot of success recently. It is his sense, he said, that a proof is close at hand, more than a century and half after it was first articulated. ‘‘Maybe 10 years,’’ he said.

    It was dinnertime when I headed to Tao’s home, a white-and-tan five-­bedroom on the western edge of campus. Tao was originally going to take his 12-year-old son, William, to a piano lesson, but William had received a callback for a Go-Gurt commercial. (He has already been in a Honda ad, in which he played the role of ‘‘boy who sleeps contentedly in the back seat.’’) While Tao’s wife, Laura, ferried William home, their daughter, Maddy, 4, finished her meal at an island in their spacious kitchen. She took a bite of her dessert — a cronut — and then clambered down her stool and began running from room to room, arms raised, squealing with delight.

    Tao has emerged as one of the field’s great bridge-­builders. At the time of his Fields Medal, he had already made discoveries with more than 30 different collaborators. Since then, he has also become a prolific math blogger with a decidedly non-­Gaussian ebullience: He celebrates the work of others, shares favorite tricks, documents his progress and delights at any corrections that follow in the comments. He has organized cooperative online efforts to work on problems. ‘‘Terry is what a great 21st-­century mathematician looks like,’’ Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison, who has collaborated with Tao, told me. He is ‘‘part of a network, always communicating, always connecting what he is doing with what other people are doing.’’

    In my visit with Tao, I noticed only one way in which he conforms to the math-­professor stereotype: an absent-mindedness that dates to his childhood. When he was a boy, he constantly lost books, even his book bag; he put clothes on backward or inside out, or he neglected to put on both socks. (This is why he wears Birkenstocks now. ‘‘One less step,’’ he explained.) As he showed me around the house, his gait was a bit awkward, as if, at some level, he was just not that interested in walking. I asked to see his office, and he pointed out an unremarkable chamber off a back hallway. He doesn’t get as much done there as he used to, he said; recently, he has been most productive on flights, when he has a block of hours away from email and all the people who hope for an audience with him.

    After William arrived home, with Laura trailing behind, we sat down for dinner: pork chops in tomato sauce, a recipe taken from a handwritten collection, its notebook cover emblazoned with a teddy bear, that Laura received as a gift from Tao’s mother. William was gregarious. The Go-Gurt callback went well. (He eventually got the part.) William has some of his father’s natural facility for mathematics — as a sixth grader, he took an online course in precalculus — but his real passions at the moment are writing, particularly fantasy, and acting, particularly improv. He was also heavy into Minecraft, though he was annoyed because he was having trouble updating his hacks. Once, he said, he and a friend tried to hack math itself by proving that 1 equals 0, but then realized that it is forbidden to divide by 0. Tao rolled his eyes.

    An effort to prove that 1 equals 0 is not likely to yield much fruit, it’s true, but the hacker’s mind-set can be extremely useful when doing math. Long ago, mathematicians invented a number that when multiplied by itself equals negative 1, an idea that seemed to break the basic rules of multiplication. It was so far outside what mathematicians were doing at the time that they called it ‘‘imaginary.’’ Yet imaginary numbers proved a powerful invention, and modern physics and engineering could not function without them.

    Early encounters with math can be misleading. The subject seems to be about learning rules — how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act. Lore has it that when David Hilbert, arguably the most influential mathematician of fin de siècle Europe, heard that a colleague had left to pursue fiction, he quipped: ‘‘He did not have enough imagination for mathematics.’’

    Math traffics in abstractions — the idea, for example, that two apples and two oranges have something in common — but much of Tao’s work has a tangible aspect. He is drawn to waves of fluid or light, or things that can be counted, or geometries that you might hold in your mind. When a question does not initially appear in such a way, he strives to transform it. Early in his career, he struggled with a problem that involved waves rotating on top of one another. He wanted to come up with a moving coordinate system that would make things easier to see, something like a virtual Steadi­cam. So he lay down on the floor and rolled back and forth, trying to see it in his mind’s eye. ‘‘My aunt caught me doing this,’’ Tao told me, laughing, ‘‘and I couldn’t explain what I was doing.’’

    Tao’s most recent work in exploding water began when a professor from Kazakhstan claimed to have completed a Navier-­Stokes proof. After looking at it, Tao felt sure that the proof was incorrect, but he decided to take this intuition a step further and show that any proof using the professor’s approach was sure to fail. While he was wading through the proof, asking colleagues for help in translating the explanatory text from the original Russian, he struck upon the notion of his imaginary, self-­replicating water contraption — drawing on ideas from engineering to make progress on a question in pure mathematics.

    The feat is as much psychological as mathematical. Many people think that substantial progress on Navier-­Stokes may be impossible, and years ago, Tao told me, he wrote a blog post concurring with this view. Now he has some small bit of hope. The twin-prime conjecture had the same feel, a sense of breaking through the wall of intimidation that has scared off many aspirants. Outside the world of mathematics, both Navier-­Stokes and the twin-prime conjecture are described as problems. But for Tao and others in the field, they are more like opponents. Tao’s opponent has been known to taunt him, convincing him that he is overlooking the obvious, or to fight back, making quick escapes when none should be possible. Now the opponent appears to have revealed a weakness. But Tao said he has been here before, thinking he has found a way through the defenses, when in fact he was being led into an ambush. ‘‘You learn to get suspicious,’’ Tao said. ‘‘You learn to be on the lookout.’’

    This is the thrill of it, and the dread. There is a shifting beneath the ground. The game is afoot.

    See the full article here.

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