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  • richardmitnick 11:18 am on October 30, 2019 Permalink | Reply
    Tags: "Nature can help solve optimization problems", , , , Mathematics,   

    From MIT News: “Nature can help solve optimization problems” 

    MIT News

    From MIT News

    October 28, 2019
    Kylie Foy | Lincoln Laboratory

    An analog circuit solves combinatorial optimization problems by using oscillators’ natural tendency to synchronize. The technology could scale up to solve these problems faster than digital computers. Image: Bryan Mastergeorge

    A low-cost analog circuit based on synchronizing oscillators could scale up quickly and cheaply to beat out digital computers.

    Today’s best digital computers still struggle to solve, in a practical time frame, a certain class of problem: combinatorial optimization problems, or those that involve combing through large sets of possibilities to find the best solution. Quantum computers hold potential to take on these problems, but scaling up the number of quantum bits in these systems remains a hurdle.

    Now, MIT Lincoln Laboratory researchers have demonstrated an alternative, analog-based way to accelerate the computing of these problems. “Our computer works by ‘computing with physics’ and uses nature itself to help solve these tough optimization problems,” says Jeffrey Chou, co-lead author of a paper about this work published in Nature’s Scientific Reports. “It’s made of standard electronic components, allowing us to scale our computer quickly and cheaply by leveraging the existing microchip industry.”

    Perhaps the most well-known combinatorial optimization problem is that of the traveling salesperson. The problem asks to find the shortest route a salesperson can take through a number of cities, starting and ending at the same one. It may seem simple with only a few cities, but the problem becomes exponentially difficult to solve as the number of cities grows, bogging down even the best supercomputers. Yet optimization problems need to be solved in the real world daily; the solutions are used to schedule shifts, minimize financial risk, discover drugs, plan shipments, reduce interference on wireless networks, and much more.

    “It has been known for a very long time that digital computers are fundamentally bad at solving these types of problems,” says Suraj Bramhavar, also a co-lead author. “Many of the algorithms that have been devised to find solutions have to trade off solution quality for time. Finding the absolute optimum solution winds up taking an unreasonably long time when the problem sizes grow.” Finding better solutions and doing so in dramatically less time could save industries billions of dollars. Thus, researchers have been searching for new ways to build systems designed specifically for optimization.

    Finding the beat

    Nature likes to optimize energy, or achieve goals in the most efficient and distributed manner. This principle can be witnessed in the synchrony of nature, like heart cells beating together or schools of fish moving as one. Similarly, if you set two pendulum clocks on the same surface, no matter when the individual pendula are set into motion, they will eventually be lulled into a synchronized rhythm, reaching their apex at the same time but moving in opposite directions (or out of phase). This phenomenon was first observed in 1665 by the Dutch scientist Christiaan Huygens. These clocks are an example of coupled oscillators, set up in such a way that energy can be transferred between them.

    “We’ve essentially built an electronic, programmable version of this [clock setup] using coupled nonlinear oscillators,” Chou says, showing a YouTube video of metronomes displaying a similar phenomenon. “The idea is that if you set up a system that encodes your problem’s energy landscape, then the system will naturally try to minimize the energy by synchronizing, and in doing so, will settle on the best solution. We can then read out this solution.”

    The laboratory’s prototype is a type of Ising machine, a computer based on a model in physics that describes a network of magnets, each of which have a magnetic “spin” orientation that can point only up or down. Each spin’s final orientation depends on its interaction with every other spin. The individual spin-to-spin interactions are defined with a specific coupling weight, which denotes the strength of their connection. The goal of an Ising machine is to find, given a specific coupling strength network, the correct configuration of each spin, up or down, that minimizes the overall system energy.

    But how does an Ising machine solve an optimization problem? It turns out that optimization problems can be mapped directly onto the Ising model, so that a set of a spins with certain coupling weights can represent each city and the distances between them in the traveling salesperson problem. Thus, finding the lowest-energy configuration of spins in the Ising model translates directly into the solution for the seller’s fastest route. However, solving this problem by individually checking each of the possible configurations becomes prohibitively difficult when the problems grow to even modest sizes.

    In recent years, there have been efforts to build quantum machines that map to the Ising model, the most notable of which is one from the Canadian company D-Wave Systems. These machines may offer an efficient way to search the large solution space and find the correct answer, although they operate at cryogenic temperatures.

    The laboratory’s system runs a similar search, but does so using simple electronic oscillators. Each oscillator represents a spin in the Ising model, and similarly takes on a binarized phase, where oscillators that are synchronized, or in phase, represent the “spin up” configuration and those that are out of phase represent the “spin down” configuration. To set the system up to solve an optimization problem, the problem is first mapped to the Ising model, translating it into programmable coupling weights connecting each oscillator.

    With the coupling weights programmed, the oscillators are allowed to run, like the pendulum arm of each clock being released. The system then naturally relaxes to its overall minimum energy state. Electronically reading out each oscillator’s final phase, representing “spin up” or “spin down,” presents the answer to the posed question. When the system ran against more than 2,000 random optimization problems, it came to the correct solution 98 percent of the time.

    Previously, researchers at Stanford University demonstrated an Ising machine [Science] that uses lasers and electronics to solve optimization problems. That work revealed the potential for a significant speedup over digital computing although, according to Chou, the system may be difficult and costly to scale to larger sizes. The goal of finding a simpler alternative ignited the laboratory’s research.

    Scaling up

    The individual oscillator circuit the team used in their demonstration is similar to circuitry found inside cellphones or Wi-Fi routers. One addition they’ve made is a crossbar architecture that allows all of the oscillators in the circuit to be directly coupled to each other. “We have found an architecture that is both scalable to manufacture and can enable full connectivity to thousands of oscillators,” Chou says. A fully connected system allows it to easily be mapped to a wide variety of optimization problems.

    “This work from Lincoln Laboratory makes innovative use of a crossbar architecture in its construction of an analog-electronic Ising machine,” says Peter McMahon, an assistant professor of applied and engineering physics at Cornell University who was not involved in this research. “It will be interesting to see how future developments of this architecture and platform perform.”

    The laboratory’s prototype Ising machine uses four oscillators. The team is now working out a plan to scale the prototype to larger numbers of oscillators, or “nodes,” and fabricate it on a printed circuit board. “If we can get to, say, 500 nodes, there is a chance we can start to compete with existing computers, and at 1,000 nodes we might be able to beat them,” Bramhavar says.

    The team sees a clear path forward to scaling up because the technology is based on standard electronic components. It’s also extremely cheap. All the parts for their prototype can be found in a typical undergraduate electrical engineering lab and were bought online for about $20.

    “What excites me is the simplicity,” Bramhavar adds. “Quantum computers are expected to demonstrate amazing performance, but the scientific and engineering challenges required to scale them up are quite hard. Demonstrating even a small fraction of the performance gains envisioned with quantum computers, but doing so using hardware from the existing electronics industry, would be a huge leap forward. Exploiting the natural behavior of these circuits to solve real problems presents a very compelling alternative for what the next era of computing could be.”

    See the full article here .

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  • richardmitnick 11:04 am on September 11, 2019 Permalink | Reply
    Tags: "The answer to life, 32 is unsolvable., 65-year-old problem about 42, and everything", Andrew Booker of Bristol University, Are there three cubes whose sum is 42?, Diophantine Equation x3+y3+z3=k, Drew Sutherland, Mathematics, ,   

    From MIT News: “The answer to life, the universe, and everything” 

    MIT News

    From MIT News

    September 10, 2019
    Sandi Miller | Department of Mathematics

    MIT mathematician Andrew “Drew” Sutherland solved a 65-year-old problem about 42. Image: Department of Mathematics

    This plot by Andrew Sutherland depicts the computation times for each of the 400,000-plus jobs that his team ran on Charity Engine’s compute grid. Each job was assigned a range of the parameter d = [x+y|, which must be a divisor of |z^3-42| for any integer solution to x^3 + y^3 + z^3 = 42. Each dot in the plot represents 25 jobs plotted according to their median runtime, with purple dots representing “smooth” values of d (those with no large prime divisors), and blue dots representing non-smooth values of d — the algorithm handles these two cases differently.
    Image: Andrew Sutherland

    Mathematics researcher Drew Sutherland helps solve decades-old sum-of-three-cubes puzzle, with help from “The Hitchhiker’s Guide to the Galaxy.”

    A team led by Andrew Sutherland of MIT and Andrew Booker of Bristol University has solved the final piece of a famous 65-year old math puzzle with an answer for the most elusive number of all: 42.

    The number 42 is especially significant to fans of science fiction novelist Douglas Adams’ “The Hitchhiker’s Guide to the Galaxy,” because that number is the answer given by a supercomputer to “the Ultimate Question of Life, the Universe, and Everything.”

    Booker also wanted to know the answer to 42. That is, are there three cubes whose sum is 42?

    This sum of three cubes puzzle, first set in 1954 at the University of Cambridge and known as the Diophantine Equation x3+y3+z3=k, challenged mathematicians to find solutions for numbers 1-100. With smaller numbers, this type of equation is easier to solve: for example, 29 could be written as 33 + 13 + 13, while 32 is unsolvable. All were eventually solved, or proved unsolvable, using various techniques and supercomputers, except for two numbers: 33 and 42.

    Booker devised an ingenious algorithm and spent weeks on his university’s supercomputer when he recently came up with a solution for 33. But when he turned to solve for 42, Booker found that the computing needed was an order of magnitude higher and might be beyond his supercomputer’s capability. Booker says he received many offers of help to find the answer, but instead he turned to his friend Andrew “Drew” Sutherland, a principal research scientist in the Department of Mathematics. “He’s a world’s expert at this sort of thing,” Booker says.

    Sutherland, whose specialty includes massively parallel computations, broke the record in 2017 for the largest Compute Engine cluster, with 580,000 cores on Preemptible Virtual Machines, the largest known high-performance computing cluster to run in the public cloud.

    Like other computational number theorists who work in arithmetic geometry, he was aware of the “sum of three cubes” problem. And the two had worked together before, helping to build the L-functions and Modular Forms Database(LMFDB), an online atlas of mathematical objects related to what is known as the Langlands Program. “I was thrilled when Andy asked me to join him on this project,” says Sutherland.

    Booker and Sutherland discussed the algorithmic strategy to be used in the search for a solution to 42. As Booker found with his solution to 33, they knew they didn’t have to resort to trying all of the possibilities for x, y, and z.

    “There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion).

    Otherwise, the main difference between the search for 33 and the search for 42 would be the size of the search and the computer platform used. Thanks to a generous offer from UK-based Charity Engine, Booker and Sutherland were able to tap into the computing power from over 400,000 volunteers’ home PCs, all around the world, each of which was assigned a range of values for d. The computation on each PC runs in the background so the owner can still use their PC for other tasks.

    Sutherland is also a fan of Douglas Adams, so the project was irresistible.

    The method of using Charity Engine is similar to part of the plot surrounding the number 42 in the “Hitchhiker” novel: After Deep Thought’s answer of 42 proves unsatisfying to the scientists, who don’t know the question it is meant to answer, the supercomputer decides to compute the Ultimate Question by building a supercomputer powered by Earth … in other words, employing a worldwide massively parallel computation platform.

    “This is another reason I really liked running this computation on Charity Engine — we actually did use a planetary-scale computer to settle a longstanding open question whose answer is 42.”

    They ran a number of computations at a lower capacity to test both their code and the Charity Engine network. They then used a number of optimizations and adaptations to make the code better suited for a massively distributed computation, compared to a computation run on a single supercomputer, says Sutherland.

    Why couldn’t Bristol’s supercomputer solve this problem?

    “Well, any computer *can* solve the problem, provided you are willing to wait long enough, but with roughly half a million PCs working on the problem in parallel (each with multiple cores), we were able to complete the computation much more quickly than we could have using the Bristol machine (or any of the machines here at MIT),” says Sutherland.

    Using the Charity Engine network is also more energy-efficient. “For the most part, we are using computational resources that would otherwise go to waste,” says Sutherland. “When you’re sitting at your computer reading an email or working on a spreadsheet, you are using only a tiny fraction of the CPU resource available, and the Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BOINC), takes advantage of this. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.”

    Sutherland and Booker ran the computations over several months, but the final successful run was completed in just a few weeks. When the email from Charity Engine arrived, it provided the first solution to x3+y3+z3=42:

    42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

    “When I heard the news, it was definitely a fist-pump moment,” says Sutherland. “With these large-scale computations you pour a lot of time and energy into optimizing the implementation, tweaking the parameters, and then testing and retesting the code over weeks and months, never really knowing if all the effort is going to pay off, so it is extremely satisfying when it does.”

    Booker and Sutherland say there are 10 more numbers, from 101-1000, left to be solved, with the next number being 114.

    But both are more interested in a simpler but computationally more challenging puzzle: whether there are more answers for the sum of three cubes for 3.

    “There are four very easy solutions that were known to the mathematician Louis J. Mordell, who famously wrote in 1953, ‘I do not know anything about the integer solutions of x3 + y3 + z3 = 3 beyond the existence of the four triples (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4); and it must be very difficult indeed to find out anything about any other solutions.’ This quote motivated a lot of the interest in the sum of three cubes problem, and the case k=3 in particular. While it is conjectured that there should be infinitely many solutions, despite more than 65 years of searching we know only the easy solutions that were already known to Mordell. It would be very exciting to find another solution for k=3.”

    See the full article here .

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  • richardmitnick 11:43 am on September 7, 2019 Permalink | Reply
    Tags: A resulting gravito-magnetic field analogous to the magnetic field surrounding the two poles of a magnet would explain the alignment of the jets with the source’s north-south axis of rotation., Albert Einstein’s equations of gravity and James Clerk Maxwell’s equations of electromagnetism., Astronomers expect that a new satellite (LARES 2) to be launched at the end of 2019 will with data from LAGEOS give an accuracy of 0.2%., Astrophysicists have already taken gravito-magnetism on board., , But how far can such mathematical analogies be pushed? Is “gravito-magnetic induction” real? If it is it should show up as a tiny wobble in the orbit of satellites., , For frame-dragging the best agreement with GR has been within 0.2%, Gravito-electromagnetism, In some ways mathematics is like literature. It has its own definitions and grammatical rules – although unfortunately these are the bane of too many students’ lives., It suggests a mechanism to explain the mysterious jets of gas that have been observed spewing out of quasars and active galactic nuclei., Making physical analogies is fundamental in the process of physics because it helps physicists to imagine new physical phenomena., Mathematics, , Rotating supermassive black holes at the heart of these cosmic powerhouses would produce enormous frame-dragging and geodetic effects., Simpler versions that work with an accuracy of 5%., The intriguing mathematical analogy between the equations of Newtonian gravity and Coulomb’s law of electrostatics., The prediction of a new force: “gravito-magnetism”, The same is true of mathematical analogies applied to physical reality – and especially of the interplay between mathematical and physical analogies., Today this so-called “gravito-electromagnetism” or GEM for short is generally treated mathematically via the “weak field” approximation to the full GR equations – simpler versions that work   

    From COSMOS Magazine: “Introducing the amazing concept of gravito-electromagnetism” 

    Cosmos Magazine bloc

    From COSMOS Magazine

    05 September 2019
    Robyn Arianrhod

    Mathematician and poet James Clerk Maxwell. Credit SIR GODFREY KNELLER / GETTY IMAGES / (BACKGROUND) SOLA

    In some ways, mathematics is like literature. It has its own definitions and grammatical rules – although unfortunately these are the bane of too many students’ lives. Which is a great pity, because when used elegantly and clearly, mathematical language can help readers to see things in entirely new ways. Take analogies, for example. They’re obviously powerful in literature – who doesn’t thrill to a creative, well-aimed metaphor? But they can be even more powerful in mathematical physics.

    Making physical analogies is fundamental in the process of physics, because it helps physicists to imagine new physical phenomena. We still speak of the “flow” of an electric “current”, using liquid metaphors that physicists coined before they knew that electrons existed. On the other hand, the old concept of “ether” – a hypothetical light-carrying medium analogous to water or air – has long passed its use-by date. Physical analogies can be creative and useful, but sometimes they can lead one astray.

    The same is true of mathematical analogies applied to physical reality – and especially of the interplay between mathematical and physical analogies. An analogy that has tantalised mathematicians and physicists for a century, and which is still a hot if much-debated topic, is that between Albert Einstein’s equations of gravity and James Clerk Maxwell’s equations of electromagnetism. It’s led to an exciting new field of research called “gravito-electromagnetism” – and to the prediction of a new force, “gravito-magnetism”.

    Diagram regarding the confirmation of Gravitomagnetism by Gravity Probe B. Gravity Probe B Team, Stanford, NASA

    The surprising idea of comparing gravity and electromagnetism – two entirely different kinds of phenomena – began with the intriguing mathematical analogy between the equations of Newtonian gravity and Coulomb’s law of electrostatics. Both sets of equations have exactly the same inverse-square form.

    In 1913, Einstein began exploring the much more complex idea of a relativistic gravitational analogue of electromagnetic induction – an idea that was developed by Josef Lense and Hans Thirring in 1918. They used Einstein’s final theory of general relativity (GR), which was published in 1916.

    Today this so-called “gravito-electromagnetism”, or GEM for short, is generally treated mathematically via the “weak field” approximation to the full GR equations – simpler versions that work well in weak fields such as that of the earth.

    It turns out that the mathematics of weak fields includes quantities satisfying equations that look remarkably similar to Maxwell’s. The “gravito-electric” part can be readily identified with the everyday Newtonian downward force that keeps us anchored to the earth. The “gravito-magnetic” part, however, is something entirely unfamiliar – a new force apparently due to the rotation of the earth (or any large mass).

    It’s analogous to the way a spinning electron produces a magnetic field via electromagnetic induction, except that mathematically, a massive spinning object mathematically “induces” a “dragging” of space-time itself – as if space-time were like a viscous fluid that’s dragged around a rotating ball. (Einstein first identified “frame-dragging”, a consequence of general relativity elaborated by Lense and Thirring.)

    But how far can such mathematical analogies be pushed? Is “gravito-magnetic induction” real? If it is, it should show up as a tiny wobble in the orbit of satellites, and – thanks also to the “geodetic” effect, the curving of space-time by matter – as a change in the direction of the axis of an orbiting gyroscope. (The latter is analogous to the way a magnetic field generated by an electric current changes the orientation of a magnetic dipole.)

    Finally, after a century of speculation, answers are unfolding. Independent results from several satellite missions – notably Gravity Probe B, LAGEOS, LARES, and GRACE – have confirmed the earth’s geodetic and frame-dragging effects to varying degrees of precision.

    NASA/Gravity Probe B

    LAGEOS satellite, courtesy of NASA

    The LARES Satellite. Italian Space Agency

    NASA/ German Research Centre for Geosciences (GFZ) Grace-FO satellites

    For frame-dragging, the best agreement with GR has been within 0.2%, with an accuracy of 5%, but astronomers expect that a new satellite (LARES 2), to be launched at the end of 2019, will, with data from LAGEOS, give an accuracy of 0.2%.

    More accurate results will provide more stringent tests of GR, but astrophysicists have already taken gravito-magnetism on board. For instance, it suggests a mechanism to explain the mysterious jets of gas that have been observed spewing out of quasars and active galactic nuclei. Rotating supermassive black holes at the heart of these cosmic powerhouses would produce enormous frame-dragging and geodetic effects. A resulting gravito-magnetic field analogous to the magnetic field surrounding the two poles of a magnet would explain the alignment of the jets with the source’s north-south axis of rotation.

    Making analogies is a tricky business, however, and there are some interpretive anomalies still to unravel. To take just one example, questions remain about the meaning of analogical terms such as gravitational “energy density” and “energy current density”. Things are perhaps even more problematic – or interesting – from the mathematical point of view.

    For example, there is another, purely mathematical analogy between Einstein’s and Maxwell’s equations, which gives rise to a very different analogy from the GEM equations. To put it briefly, it’s a comparison between the so-called Bianchi identities in each theory.

    The existence of two (and in fact several) such different mathematical analogies between the equations of these two physical phenomena is incredibly suggestive of a deeper connection. At present, though, there are some apparent physical inconsistencies between the “electric” and “magnetic” parts in each mathematical approach.

    Still, the formal analogies are useful in helping mathematicians find intuitively familiar ways to think about the formidable equations of GR. And there’s always the tantalising possibility that this approach will prove as physically profound as the prediction of gravito-magnetism.

    See the full article here .

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  • richardmitnick 12:27 pm on July 28, 2019 Permalink | Reply
    Tags: “Patience is important for our subject” says math professor Wei Zhang. “You’re always making infinitesimal progress.”, He was focusing on L-functions- an important area of number theory., He was in the last year of his PhD studies in mathematics at Columbia University, Mathematics, , Wei Zhang, Wei Zhang’s breakthrough happened on the train. He was riding home to New York after visiting a friend in Boston.   

    From MIT News: “Mathematical insights through collaboration and perseverance” 

    MIT News

    From MIT News

    July 28, 2019
    Jonathan Mingle

    Wei Zhang. Image: Jake Belcher

    “Patience is important for our subject,” says math professor Wei Zhang. “You’re always making infinitesimal progress.”

    Wei Zhang’s breakthrough happened on the train. He was riding home to New York after visiting a friend in Boston, during the last year of his PhD studies in mathematics at Columbia University, where he was focusing on L-functions, an important area of number theory.

    “All of a sudden, things were linked together,” he recalls, about the flash of insight that allowed him to finish a key project related to his dissertation. “Definitely it was an ‘Aha!’ moment.”

    But that moment emerged from years of patient study and encounters with other mathematicians’ ideas. For example, he had attended talks by a certain faculty member in his first and third years at Columbia, but each time he thought the ideas presented in those lectures wouldn’t be relevant for his own work.

    “And then two years later, I found this was exactly what I needed to finish a piece of the project!” says Zhang, who joined MIT two years ago as a professor of mathematics.

    As Zhang recalls, during that pivotal train ride his mind had been free to wander around the problem and consider it from different angles. With this mindset, “I can have a more panoramic way of putting everything into one piece. It’s like a puzzle — when you close your eyes maybe you can see more. And when the mind is trying to organize different parts of a story, you see this missing part.”

    Allowing time for this panoramic view to come into focus has been critical throughout Zhang’s career. His breakthrough on the train 11 years ago led him to propose a set of conjectures that he has just now solved in a recent paper.

    “Patience is important for our subject,” he says. “You’re always making infinitesimal progress. All discovery seems to be made in one moment. But without the preparation and long-time accumulation of knowledge, it wouldn’t be possible.”

    An early and evolving love for math

    Zhang traces his interest in math back to the fourth grade in his village school in a remote part of China’s Sichuan Province. “It was just pure curiosity,” he says. “Some of the questions were so beautifully set up.”

    He started participating in math competitions. Seeing his potential, a fifth-grade math teacher let Zhang pore over an extracurricular book of problems. “Those questions made me wonder how such simple solutions to seemingly very complicated questions could be possible,” he says.

    Zhang left home to attend a high school 300 miles away in Chengdu, the capital city of Sichuan. By the time he applied to study at Peking University in Beijing, he knew he wanted to study mathematics. And by his final year there, he had decided to pursue a career as a mathematician.

    He credits one of his professors with awakening him to some exciting frontiers and more advanced areas of study, during his first year. At that time, around 2000, the successful proof of Fermat’s Last Theorem by Andrew Wiles five years earlier was still relatively fresh, and reverberating through the world of mathematics. “This teacher really liked to chat,” Zhang says, “and he explained the contents of some of those big events and results in a way that was accessible to first-year students.”

    “Later on, I read those texts by myself, and I found it was something I liked,” he says. “The tools being developed to prove Fermat’s Last Theorem were a starting point for me.”

    Today, Zhang gets to cultivate his own students’ passion for math, even as his teaching informs his own research. “It has happened more than once for me, that while teaching I got inspired,” he says. “For mathematicians, we may understand some sort of result, but that doesn’t mean we actually we know how to prove them. By teaching a course, it really helps us go through the whole process. This definitely helps, especially with very talented students like those at MIT.”

    From local to global information

    Zhang’s core area of research and expertise is number theory, which is devoted to the study of integers and their properties. Broadly speaking, Zhang explores how to solve equations in integers or in rational numbers. A familiar example is a Pythagorean triple (a2+b2=c2).

    “One simple idea is try to solve equations with modular arithmetic,” he says. The most common example of modular arithmetic is a 12-hour clock, which counts time by starting over and repeating after it reaches 12. With modular arithmetic, one can compile a set of data, indexed, for example, by prime numbers.

    “But after that, how do you return to the initial question?” he says. “Can you tell an equation has an integer solution by collecting data from modular arithmetic?” Zhang investigates whether and how an equation can be solved by restoring this local data to a global piece of information — like finding a Pythagorean triple.

    His research is relevant to an important facet of the Langlands Program — a set of conjectures proposed by mathematician Robert Langlands for connecting number theory and geometry, which some have likened to a kind of “grand unified theory” of mathematics.

    Conversations and patience

    Bridging other branches of math with number theory has become one of Zhang’s specialties.

    In 2018, he won the New Horizons in Mathematics Breakthroughs Prize, a prestigious award for researchers early in their careers. He shared the prize with his old friend and undergraduate classmate, and current MIT colleague, Zhiwei Yun, for their joint work [Anals of Mathematics] on the Taylor expansion of L-functions, which was hailed as a major advance in a key area of number theory in the past few decades.

    Their project grew directly out of his dissertation research. And that work, in turn, opened up new directions in his current research, related to the arithmetic of elliptic curves. But Zhang says the way forward wasn’t clear until five years — and many conversations with Yun — later.

    “Conversation is important in mathematics,” Zhang says. “Very often mathematical questions can be solved, or at least progress can be made, by bringing together people with different skills and backgrounds, with new interpretations of the same set of facts. In our case, this is a perfect example. His geometrical way of thinking about the question was exactly complementary to my own perspective, which is more number arithmetic.”

    Lately, Zhang’s work has taken place on fewer train rides and more flights. He travels back to China at least once a year, to visit family and colleagues in Beijing. And when he feels stuck on a problem, he likes to take long walks, play tennis, or simply spend time with his young children, to clear his mind.

    His recent solution of his own conjecture has led him to contemplate unexplored terrain. “This opened up a new direction,” he says. “I think it’s possible to finally get some higher-dimensional solutions. It opens up new conjectures.”

    See the full article here .

    Please help promote STEM in your local schools.

    Stem Education Coalition

    MIT Seal

    The mission of MIT is to advance knowledge and educate students in science, technology, and other areas of scholarship that will best serve the nation and the world in the twenty-first century. We seek to develop in each member of the MIT community the ability and passion to work wisely, creatively, and effectively for the betterment of humankind.

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  • richardmitnick 2:21 pm on April 16, 2019 Permalink | Reply
    Tags: , , , , Mathematics, Natural Sciences, The Brendan Iribe Center for Computer Science and Engineering, UMIACS-University of Maryland Institute for Advanced Computer Studies,   

    From University of Maryland CMNS: “University of Maryland Launches Center for Machine Learning” 

    U Maryland bloc

    From University of Maryland


    April 16, 2019

    Abby Robinson

    The University of Maryland recently launched a multidisciplinary center that uses powerful computing tools to address challenges in big data, computer vision, health care, financial transactions and more.

    The University of Maryland Center for Machine Learning will unify and enhance numerous activities in machine learning already underway on the Maryland campus.

    University of Maryland computer science faculty member Thomas Goldstein (on left, with visiting graduate student) is a member of the new Center for Machine Learning. Goldstein’s research focuses on large-scale optimization and distributed algorithms for big data. Photo: John T. Consoli.

    Machine learning uses algorithms and statistical models so that computer systems can effectively perform a task without explicit instructions, relying instead on patterns and inference. At UMD, for example, computer vision experts are “training” computers to identify and match key facial characteristics by having machines analyze millions of images publicly available on social media.

    Researchers at UMD are exploring other applications such as groundbreaking work in cancer genomics; powerful algorithms to improve the selection process for organ transplants; and an innovative system that can quickly find, translate and summarize information from almost any language in the world.

    “We wanted to capitalize on the significant strengths we already have in machine learning, provide additional support, and embrace fresh opportunities arising from new facilities and partnerships,” said Mihai Pop, professor of computer science and director of the University of Maryland Institute for Advanced Computer Studies (UMIACS).

    The center officially launched with a workshop last month featuring talks and panel discussions from machine learning experts in auditory systems, biology and medicine, business, chemistry, natural language processing, and security.

    Initial funding for the center comes from the College of Computer, Mathematical, and Natural Sciences (CMNS) and UMIACS, which will provide technical and administrative support.

    An inaugural partner of the center, financial and technology leader Capital One, provided additional support, including endowing three faculty positions in machine learning and computer science. Those positions received matching funding from the state’s Maryland E-Nnovation Initiative.

    Capital One has also provided funding for research projects that align with the organization’s need to stay on the cutting edge in areas like fraud detection and enhancing the customer experience with more personalized, real-time features.

    “We are proud to be a part of the launch of the University of Maryland Center for Machine Learning, and are thrilled to extend our partnership with the university in this field,” said Dave Castillo, the company’s managing vice president at the Center for Machine Learning and Emerging Technology. “At Capital One, we believe forward-leaning technologies like machine learning can provide our customers greater protection, security, confidence and control of their finances. We look forward to advancing breakthrough work with the University of Maryland in years to come.”

    University of Maryland computer science faculty members David Jacobs (left) and Furong Huang (right) are part of the new Center for Machine Learning. Jacobs is an expert in computer vision and is the center’s interim director; Huang is conducting research in neural networks. Photo: John T. Consoli.

    David Jacobs, a professor of computer science with an appointment in UMIACS, will serve as interim director of the new center.

    To jumpstart the center’s activities, Jacobs has recruited a core group of faculty members in computer science and UMIACS: John Dickerson, Soheil Feizi, Thomas Goldstein, Furong Huang and Aravind Srinivasan.

    Faculty members from mathematics, chemistry, biology, physics, linguistics, and data science are also heavily involved in machine learning applications, and Jacobs said he expects many of them to be active in the center through direct or affiliate appointments.

    “We want the center to be a focal point across the campus where faculty, students, and visiting scholars can come to learn about the latest technologies and theoretical applications based in machine learning,” he said.

    Key to the center’s success will be a robust computational infrastructure that is needed to perform complex computations involving massive amounts of data.

    This is where UMIACS plays an important role, Jacobs said, with the institute’s technical staff already supporting multiple machine learning activities in computer vision and computational linguistics.

    Plans call for CMNS, UMIACS and other organizations to invest substantially in new computing resources for the machine learning center, Jacobs added.

    The Brendan Iribe Center for Computer Science and Engineering. Photo: John T. Consoli.

    The center will be located in the Brendan Iribe Center for Computer Science and Engineering, a new state-of-the-art facility at the entrance to campus that will be officially dedicated later this month. In addition to the very latest in computing resources, the Brendan Iribe Center promotes collaboration and connectivity through its open design and multiple meeting areas.

    The Brendan Iribe Center is directly adjacent to the university’s Discovery District, where researchers working in Capital One’s Tech Incubator and other tech startups can interact with UMD faculty members and students on topics related to machine learning.

    Amitabh Varshney, professor of computer science and dean of CMNS, said the center will be a valuable resource for the state of Maryland and the region—both for students seeking the latest knowledge and skills and for companies wanting professional development training for their employees.

    “We have new educational activities planned by the college that include professional master’s programs in machine learning and data science and analytics,” Varshney said. “We want to leverage our location near numerous federal agencies and private corporations that are interested in expanding their workforce capabilities in these areas.”

    See the full article here .


    Please help promote STEM in your local schools.

    Stem Education Coalition

    U Maryland Campus

    About CMNS

    The thirst for new knowledge is a fundamental and defining characteristic of humankind. It is also at the heart of scientific endeavor and discovery. As we seek to understand our world, across a host of complexly interconnected phenomena and over scales of time and distance that were virtually inaccessible to us a generation ago, our discoveries shape that world. At the forefront of many of these discoveries is the College of Computer, Mathematical, and Natural Sciences (CMNS).

    CMNS is home to 12 major research institutes and centers and to 10 academic departments: astronomy, atmospheric and oceanic science, biology, cell biology and molecular genetics, chemistry and biochemistry, computer science, entomology, geology, mathematics, and physics.

    Our Faculty

    Our faculty are at the cutting edge over the full range of these disciplines. Our physicists fill in major gaps in our fundamental understanding of matter, participating in the recent Higgs boson discovery, and demonstrating the first-ever teleportation of information between atoms. Our astronomers probe the origin of the universe with one of the world’s premier radio observatories, and have just discovered water on the moon. Our computer scientists are developing the principles for guaranteed security and privacy in information systems.

    Our Research

    Driven by the pursuit of excellence, the University of Maryland has enjoyed a remarkable rise in accomplishment and reputation over the past two decades. By any measure, Maryland is now one of the nation’s preeminent public research universities and on a path to become one of the world’s best. To fulfill this promise, we must capitalize on our momentum, fully exploit our competitive advantages, and pursue ambitious goals with great discipline and entrepreneurial spirit. This promise is within reach. This strategic plan is our working agenda.

    The plan is comprehensive, bold, and action oriented. It sets forth a vision of the University as an institution unmatched in its capacity to attract talent, address the most important issues of our time, and produce the leaders of tomorrow. The plan will guide the investment of our human and material resources as we strengthen our undergraduate and graduate programs and expand research, outreach and partnerships, become a truly international center, and enhance our surrounding community.

    Our success will benefit Maryland in the near and long term, strengthen the State’s competitive capacity in a challenging and changing environment and enrich the economic, social and cultural life of the region. We will be a catalyst for progress, the State’s most valuable asset, and an indispensable contributor to the nation’s well-being. Achieving the goals of Transforming Maryland requires broad-based and sustained support from our extended community. We ask our stakeholders to join with us to make the University an institution of world-class quality with world-wide reach and unparalleled impact as it serves the people and the state of Maryland.

    Our researchers are also at the cusp of the new biology for the 21st century, with bioscience emerging as a key area in almost all CMNS disciplines. Entomologists are learning how climate change affects the behavior of insects, and earth science faculty are coupling physical and biosphere data to predict that change. Geochemists are discovering how our planet evolved to support life, and biologists and entomologists are discovering how evolutionary processes have operated in living organisms. Our biologists have learned how human generated sound affects aquatic organisms, and cell biologists and computer scientists use advanced genomics to study disease and host-pathogen interactions. Our mathematicians are modeling the spread of AIDS, while our astronomers are searching for habitable exoplanets.

    Our Education

    CMNS is also a national resource for educating and training the next generation of leaders. Many of our major programs are ranked among the top 10 of public research universities in the nation. CMNS offers every student a high-quality, innovative and cross-disciplinary educational experience that is also affordable. Strongly committed to making science and mathematics studies available to all, CMNS actively encourages and supports the recruitment and retention of women and minorities.

    Our Students

    Our students have the unique opportunity to work closely with first-class faculty in state-of-the-art labs both on and off campus, conducting real-world, high-impact research on some of the most exciting problems of modern science. 87% of our undergraduates conduct research and/or hold internships while earning their bachelor’s degree. CMNS degrees command respect around the world, and open doors to a wide variety of rewarding career options. Many students continue on to graduate school; others find challenging positions in high-tech industry or federal laboratories, and some join professions such as medicine, teaching, and law.

  • richardmitnick 2:24 pm on February 6, 2019 Permalink | Reply
    Tags: Active Learning Initiative funds nine projects, Biological and Environmental Engineering, , Ecology and Evolutionary Biology, Entomology, In all 70 faculty members will work on substantially changing the way they teach in more than 40 courses to over 4500 students. The work will be supported by 17 new teaching innovation postdoctoral fe, Information Science, Mathematics, Mechanical and Aerospace Engineering, Natural Resources, Psychology, The School of Integrative Plant Science   

    From Cornell Chronicle: “Active Learning Initiative funds nine projects” 

    Cornell Bloc

    From Cornell Chronicle

    February 6, 2019
    Daniel Aloi

    Students work together in Introduction to Evolutionary Biology and Diversity, an Active Learning Initiative course. Cornell Brand Communications File Photo.

    Cornell’s Active Learning Initiative (ALI) will nearly double in scope and impact with a new round of funding for innovative projects to enhance undergraduate teaching and learning in nine departments.

    In the first universitywide ALI grant competition, about $5 million has been awarded in substantial new grants ranging from $195,000 to almost $1 million, spread over two to five years. The funded projects will affect courses at all levels, including sequences aimed at majors, survey courses for non-majors, and introductory, online and lab courses.

    In all, 70 faculty members will work on substantially changing the way they teach in more than 40 courses to over 4,500 students. The work will be supported by 17 new teaching innovation postdoctoral fellows across the projects.

    The initiative aims to improve teaching and learning in groups of courses by introducing active learning and other research-based pedagogies drawn from a variety of disciplines. Two previous grant cycles in 2014 and 2017 focused on projects within the College of Arts and Sciences.

    Undergraduate teaching departments across the university received a call for proposals last fall. The Departments of Mathematics and of Ecology and Evolutionary Biology won their second ALI grants, and large projects in information science and engineering are among those funded this cycle.

    “We received many excellent and thoughtful proposals,” said Vice Provost for Academic Innovation Julia Thom-Levy, who supervises the initiative with support from the Center for Teaching Innovation. “Over the three competitions, we have already or will work with more than 100 faculty in 16 departments and four colleges, putting Cornell at the cutting edge of innovation in undergraduate education. This is an extremely exciting development, and many people have worked hard to get us to this point.”

    The grants have so far supported projects in the natural sciences, social sciences, engineering, mathematics and the humanities. Projects are jointly funded by ALI and the respective colleges, with support for the initiative coming from the Office of the Provost and a donor.

    The departments and projects funded:

    Information Science will transform six core courses over the next three years. Faculty and postdocs will incorporate innovative techniques for activities in and out of the classroom, including live-coding collaborations and group data visualization projects. The project explores how to facilitate student learning and implement collaborative classwork and peer feedback with increasingly large class sizes. Impact: more than 1,500 students over three years.

    Mathematics will redesign two linear algebra courses providing foundational math knowledge for many fields, with a target of improving students’ conceptual understanding and ability to model real-life situations; and the department will continue to develop instructor training. Impact: more than 400 students a year. The department received a three-year ALI grant in 2017 to transform two introductory calculus courses and a proofs course, together serving more than 900 students a year.

    Biological and Environmental Engineering: Three existing courses and one new course will focus on developing problem-solving skills that span disciplines, allowing students to transfer skills and knowledge across courses and contexts, and identify and develop solutions to complex problems. Overall impact of the three-year grant: About 200 students will take these courses every year.

    Ecology and Evolutionary Biology faculty will take active learning a step further following a five-year ALI grant in 2014 that transformed two core introductory courses. A new, online active learning version of one, Evolutionary Biology and Diversity, will launch to run parallel to the classroom course during the academic year and on its own in the summer. Goals of the three-year project include reaching a broader, more diverse group of students without increasing an already large class size; and establishing a model for designing online courses and assessing their effectiveness in comparison to the in-person course that is already offered on campus.

    Entomology faculty will redesign three popular classes for non-majors with a three-year grant. Active learning modules will be incorporated to prompt students to practice thinking and communicating like scientists, and learn to critically evaluate and interpret scientific information. Impact: more than 300 students a year.

    Mechanical and Aerospace Engineering faculty have developed a plan to transform six courses and combine the best elements of project teams and coursework through case-based learning. The courses are taken simultaneously by nearly all MAE students as juniors, allowing for projects and assignments spanning multiple courses, focusing on different aspects of the same engineering challenge. Impact of the project, funded by a four-year grant: “a richer and more applied engineering experience” for more than 130 students a year.

    The School of Integrative Plant Science plans to further transform its core 10-course undergraduate curriculum with a five-year grant. SIPS revised the curriculum when it was established in 2015, and enrollment in the major has since more than doubled in size. The grant will support the work of 14 faculty members and four postdocs, developing in-class activities to improve student learning and targeting the laboratory components of the program by moving away from observational labs and toward experimental labs.

    Natural Resources: Faculty teaching in the multidisciplinary Environmental and Sustainability Sciences (ESS) major will redesign an online course on Climate Solutions and a Field Biology course, and develop new courses aimed at collaboratively solving complex environmental problems, such as improving water resource management and assessing environmental policy. Climate Solutions students on campus can engage in discussions with students from around the world taking a parallel MOOC version of the class. Natural resources faculty will lead these efforts over three years; the rapidly growing ESS major involves 75 faculty members from 22 departments across the Colleges of Agriculture and Life Sciences and Arts and Sciences.

    Psychology: Introduction to Psychology, one of the largest courses at Cornell with more than 800 students, will be transformed as part of an ALI-funded, three-year project to implement active learning strategies in several undergraduate courses. Faculty aim to introduce polling questions and student discussion in the large course and more inquiry-driven group work in smaller classes. The project will target learning outcomes established by the American Psychological Association.

    “The new projects build on impressive results from previous competitions within the College of Arts and Sciences,” said Peter Lepage, the Goldwin Smith Professor of Physics and director of ALI. “Research shows that student learning can be improved dramatically through active learning, and that is what we are finding at Cornell.”

    ALI, together with the Center for Teaching Innovation, works with departments throughout the grant period, helps train staff in active learning and helps departments design assessments to measure impacts.

    See the full article here .


    Please help promote STEM in your local schools.

    Stem Education Coalition

    Once called “the first American university” by educational historian Frederick Rudolph, Cornell University represents a distinctive mix of eminent scholarship and democratic ideals. Adding practical subjects to the classics and admitting qualified students regardless of nationality, race, social circumstance, gender, or religion was quite a departure when Cornell was founded in 1865.

    Today’s Cornell reflects this heritage of egalitarian excellence. It is home to the nation’s first colleges devoted to hotel administration, industrial and labor relations, and veterinary medicine. Both a private university and the land-grant institution of New York State, Cornell University is the most educationally diverse member of the Ivy League.

    On the Ithaca campus alone nearly 20,000 students representing every state and 120 countries choose from among 4,000 courses in 11 undergraduate, graduate, and professional schools. Many undergraduates participate in a wide range of interdisciplinary programs, play meaningful roles in original research, and study in Cornell programs in Washington, New York City, and the world over.

  • richardmitnick 2:57 pm on May 8, 2018 Permalink | Reply
    Tags: , , , , Mathematics, , ,   

    From Symmetry: “Leveling the playing field” 

    Symmetry Mag
    From Symmetry

    Photo by Eleanor Starkman

    Ali Sundermier

    [When I read this article, my first reaction was that this is all worthless. I have been running a series in this blog which highlights “Women in STEM” in all of the phases that the expression implies. The simple fact is that there is and continues to be and will continue to be gender bias in the physical sciences (and probably elsewhere, but this is my area of choice). This is certainly unfair to women, but it is also unfair to all of mankind. We are losing a lot of great and powerful minds and voices as we try to push the future of knowledge and quality of life for all. So I am doing the post. But in all fields men need to call on and respect women if things are to improve. I personally see no evidence of this. As long as women only get to talk to women there will be no progress.]

    Conferences for Undergraduate Women in Physics aims to encourage more women and gender minorities to pursue careers in physics and improve diversity in the field.

    Nicole Pfiester, an engineering grad student at Tufts University, says she has been interested in physics since she was a child. She says she loves learning how things work, and physics provides a foundation for doing just that.

    But when Pfiester began pursuing a degree in physics as an undergraduate at Purdue University in 2006, she had a hard time feeling like she belonged in the male-dominated field.

    “In a class of about 30 physics students,” she says, “I think two of us were women. I just always stood out. I was kind of shy back then and much more inclined to open up to other women than I was to men, especially in study groups. Not being around people I could relate to, while it didn’t make things impossible, definitely made things more difficult.”

    In 2008, two years into her undergraduate career, Pfiester attended a conference at the University of Michigan that was designed to address this very issue. The meeting was part of the Conferences for Undergraduate Women in Physics, or CUWiP, a collection of annual three-day regional conferences to give undergraduate women a sense of belonging and motivate them to continue in the field.

    Pfiester says it was amazing to see so many female physicists in the same room and to learn that they had all gone through similar experiences. It inspired her and the other students she was with to start their own Women in Physics chapter at Purdue. Since then, the school has hosted two separate CUWiP events, in 2011 and 2015.

    “Just seeing that there are other people like you doing what it is you want to do is really powerful,” Pfiester says. “It can really help you get through some difficult moments where it’s really easy, especially in college, to feel like you don’t belong. When you see other people experiencing the same struggles and, even more importantly, you see role models who look and talk like you, you realize that this is something you can do, too. I always left those conferences really energized and ready to get back into it.”

    CUWiP was founded in 2006 when two graduate students at the University of Southern California realized that only 21 percent of US undergraduates in physics were women, a percentage that dropped even further in physics with career progression. In the 12 years since then, the percentage of undergraduate physics degrees going to women in the US has not grown, but CUWiP has. What began as one conference with 27 attendees has developed into a string of conferences held at sites across the country, as well as in Canada and the UK, with more than 1500 attendees per year. Since the American Physical Society took the conference under its umbrella in 2012, the number of participants has continued to grow every year.

    The conferences are supported by the National Science Foundation, the Department of Energy and the host institutions. Most student transportation to the conferences is almost covered by the students’ home institutions, and APS provides extensive administrative support. In addition, local organizing committees contribute a significant volunteer effort.

    “We want to provide women, gender minorities and anyone who attends the conference access to information and resources that are going to help them continue in science careers,” says Pearl Sandick, a dark-matter physicist at the University of Utah and chair of the National Organizing Committee for CUWiP.

    Some of the goals of the conference, Sandick says, are to make sure people leave with a greater sense of community, identify themselves more as physicists, become more aware of gender issues in physics, and feel valued and respected in their field. They accomplish this through workshops and panels featuring accomplished female physicists in a broad range of professions.

    Before the beginning of the shared video keynote talk, attendees at each CUWiP site cheer and wave on video. This gives a sense of the national scale of the conference and the huge number of people involved.
    Courtesy of Columbia University

    “Often students come to the conference and are very discouraged,” says past chair Daniela Bortoletto, a high-energy physicist at the University of Oxford who organizes CUWiP in the UK. “But then they meet these extremely accomplished scientists who tell the stories of their lives, and they learn that everybody struggles at different steps, everybody gets discouraged at some point, and there are ups and downs in everyone’s careers. I think it’s valuable to see that. The students walk out of the conference with a lot more confidence.”

    Through CUWiP, the organizers hope to equip students to make informed decisions about their education and expose them to the kinds of career opportunities that are open to them as physics majors, whether it means going to grad school or going into industry or science policy.

    “Not every student in physics is aware that physicists do all kinds of things,” says Kate Scholberg, a neutrino physicist at Duke and past chair. “Everybody who has been a physics undergrad gets the question, ‘What are you going to do with that?’ We want to show students there’s a lot more out there than grad school and help them expand their professional networks.”

    They also reach back to try to make conditions better for the next generations of physicists.

    At the 2018 conference, Hope Marks, now a senior at Utah State University majoring in physics, participated in a workshop in which she wrote a letter to her high school physics teacher, who she says really sparked her interest in the field.

    “I really liked the experiments we did and talking about some of the modern discoveries of physics,” she says. “I loved how physics allows us to explore the world from particles even smaller than atoms to literally the entire universe.”

    The workshop was meant to encourage high school science and math teachers to support women in their classes.

    One of the challenges to organizing the conferences, says Pat Burchat, an observational cosmologist at Stanford University and past chair, is to build experiences that are engaging and accessible for undergraduate women.

    “The tendency of organizers is naturally to think about the kinds of conferences they go to,” says Burchat says, “which usually consist of a bunch of research talks, often full of people sitting passively listening to someone talk. We want to make sure CUWiP consists of a lot of interactive sessions and workshops to keep the students engaged.”

    Candace Bryan, a physics major at the University of Utah who has wanted to be an astronomer since elementary school, says one of the most encouraging parts of the conference was learning about imposter syndrome, which occurs when someone believes that they have made it to where they are only by chance and don’t feel deserving of their achievements.

    “Science can be such an intimidating field,” she says. “It was the first time I’d ever heard that phrase, and it was really freeing to hear about it and know that so many others felt the same way. Every single person in that room raised their hand when they asked, ‘Who here has experienced imposter syndrome?’ That was really powerful. It helped me to try to move past that and improve awareness.”

    Women feeling imposter syndrome sometimes interpret their struggles as a sign that they don’t belong in physics, Bryan says.

    “It’s important to support women in physics and make sure they know there are other women out there who are struggling with the same things,” she says.

    “It was really inspirational for everyone to see how far they had come and receive encouragement to keep going. It was really nice to have that feeling after conference of ‘I can go to that class and kill it,’ or ‘I can take that test and not feel like I’m going to fail.’ And if you do fail, it’s OK, because everyone else has at some point. The important thing is to keep going.”

    See the full article here .

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    Symmetry is a joint Fermilab/SLAC publication.

  • richardmitnick 7:44 pm on April 24, 2018 Permalink | Reply
    Tags: , “Newton was the first physicist” says Sylvester James Gates a physicist at Brown University, Mathematics, Peter Woit - “When you go far enough back you really can’t tell who’s a physicist and who’s a mathematician”, , Riemannian geometry, , The relationship between physics and mathematics goes back to the beginning of both subjects   

    From Symmetry: “The coevolution of physics and math” 

    Symmetry Mag

    Evelyn Lamb

    Artwork by Sandbox Studio, Chicago

    Breakthroughs in physics sometimes require an assist from the field of mathematics—and vice versa.

    In 1912, Albert Einstein, then a 33-year-old theoretical physicist at the Eidgenössische Technische Hochschule in Zürich, was in the midst of developing an extension to his theory of special relativity.

    With special relativity, he had codified the relationship between the dimensions of space and time. Now, seven years later, he was trying to incorporate into his theory the effects of gravity. This feat—a revolution in physics that would supplant Isaac Newton’s law of universal gravitation and result in Einstein’s theory of general relativity—would require some new ideas.

    Fortunately, Einstein’s friend and collaborator Marcel Grossmann swooped in like a waiter bearing an exotic, appetizing delight (at least in a mathematician’s overactive imagination): Riemannian geometry.

    This mathematical framework, developed in the mid-19th century by German mathematician Bernhard Riemann, was something of a revolution itself. It represented a shift in mathematical thinking from viewing mathematical shapes as subsets of the three-dimensional space they lived in to thinking about their properties intrinsically. For example, a sphere can be described as the set of points in 3-dimensional space that lie exactly 1 unit away from a central point. But it can also be described as a 2-dimensional object that has particular curvature properties at every single point. This alternative definition isn’t terribly important for understanding the sphere itself but ends up being very useful with more complicated manifolds or higher-dimensional spaces.

    By Einstein’s time, the theory was still new enough that it hadn’t completely permeated through mathematics, but it happened to be exactly what Einstein needed. Riemannian geometry gave him the foundation he needed to formulate the precise equations of general relativity. Einstein and Grossmann were able to publish their work later that year.

    “It’s hard to imagine how he would have come up with relativity without help from mathematicians,” says Peter Woit, a theoretical physicist in the Mathematics Department at Columbia University.

    The story of general relativity could go to mathematicians’ heads. Here mathematics seems to be a benevolent patron, blessing the benighted world of physics with just the right equations at the right time.

    But of course the interplay between mathematics and physics is much more complicated than that. They weren’t even separate disciplines for most of recorded history. Ancient Greek, Egyptian and Babylonian mathematics took as an assumption the fact that we live in a world in which distance, time and gravity behave in a certain way.

    “Newton was the first physicist,” says Sylvester James Gates, a physicist at Brown University. “In order to reach the pinnacle, he had to invent a new piece of mathematics; it’s called calculus.”

    Calculus made some classical geometry problems easier to solve, but its foremost purpose to Newton was to give him a way to analyze the motion and change he observed in physics. In that story, mathematics is perhaps more of a butler, hired to help keep the affairs in order, than a savior.

    Even after physics and mathematics began their separate evolutionary paths, the disciplines were closely linked. “When you go far enough back, you really can’t tell who’s a physicist and who’s a mathematician,” Woit says. (As a mathematician, I was a bit scandalized the first time I saw Emmy Noether’s name attached to physics! I knew her primarily through abstract algebra.)

    Throughout the history of the two fields, mathematics and physics have each contributed important ideas to the other. Mathematician Hermann Weyl’s work on mathematical objects called Lie groups provided an important basis for understanding symmetry in quantum mechanics. In his 1930 book The Principles of Quantum Mechanics, theoretical physicist Paul Dirac introduced the Dirac delta function to help describe the concept in particle physics of a pointlike particle—anything so small that it would be modeled by a point in an idealized situation. A picture of the Dirac delta function looks like a horizontal line lying along the bottom of the x axis of a graph, at x=0, except at the place where it intersects with the y axis, where it explodes into a line pointing up to infinity. Dirac declared that the integral of this function, the measure of the area underneath it, was equal to 1. Strictly speaking, no such function exists, but Dirac’s use of the Dirac delta eventually spurred mathematician Laurent Schwartz to develop the theory of distributions in a mathematically rigorous way. Today distributions are extraordinarily useful in the mathematical fields of ordinary and partial differential equations.

    Though modern researchers focus their work more and more tightly, the line between physics and mathematics is still a blurry one. A physicist has won the Fields Medal, one of the most prestigious accolades in mathematics. And a mathematician, Maxim Kontsevich, has won the new Breakthrough Prizes in both mathematics and physics. One can attend seminar talks about quantum field theory, black holes, and string theory in both math and physics departments. Since 2011, the annual String Math conference has brought mathematicians and physicists together to work on the intersection of their fields in string theory and quantum field theory.

    String theory is perhaps the best recent example of the interplay between mathematics and physics, for reasons that eventually bring us back to Einstein and the question of gravity.

    String theory is a theoretical framework in which those pointlike particles Dirac was describing become one-dimensional objects called strings. Part of the theoretical model for those strings corresponds to gravitons, theoretical particles that carry the force of gravity.

    Most humans will tell you that we perceive the universe as having three spatial dimensions and one dimension of time. But string theory naturally lives in 10 dimensions. In 1984, as the number of physicists working on string theory ballooned, a group of researchers including Edward Witten, the physicist who was later awarded a Fields Medal, discovered that the extra six dimensions of string theory needed to be part of a space known as a Calabi-Yau manifold.

    When mathematicians joined the fray to try to figure out what structures these manifolds could have, physicists were hoping for just a few candidates. Instead, they found boatloads of Calabi-Yaus. Mathematicians still have not finished classifying them. They haven’t even determined whether their classification has a finite number of pieces.

    As mathematicians and physicists studied these spaces, they discovered an interesting duality between Calabi-Yau manifolds. Two manifolds that seem completely different can end up describing the same physics. This idea, called mirror symmetry, has blossomed in mathematics, leading to entire new research avenues. The framework of string theory has almost become a playground for mathematicians, yielding countless new avenues of exploration.

    Mina Aganagic, a theoretical physicist at the University of California, Berkeley, believes string theory and related topics will continue to provide these connections between physics and math.

    “In some sense, we’ve explored a very small part of string theory and a very small number of its predictions,” she says. Mathematicians and their focus on detailed rigorous proofs bring one point of view to the field, and physicists, with their tendency to prioritize intuitive understanding, bring another. “That’s what makes the relationship so satisfying.”

    The relationship between physics and mathematics goes back to the beginning of both subjects; as the fields have advanced, this relationship has gotten more and more tangled, a complicated tapestry. There is seemingly no end to the places where a well-placed set of tools for making calculations could help physicists, or where a probing question from physics could inspire mathematicians to create entirely new mathematical objects or theories.

    See the full article here .

    Please help promote STEM in your local schools.

    STEM Icon

    Stem Education Coalition

    Symmetry is a joint Fermilab/SLAC publication.

  • richardmitnick 12:12 pm on January 25, 2018 Permalink | Reply
    Tags: , , , Mathematicians work to expand their new pictorial mathematical language into other areas, Mathematics, Picture-perfect approach to science   

    From Harvard Gazette: “Picture-perfect approach to science” 

    Harvard University
    Harvard University

    Harvard Gazette

    January 24, 2018
    Peter Reuell

    Zhengwei Liu (left) and Arthur Jaffe are leading a new project to expand quon, their pictorial math language developed to help understand quantum information theory, into new fields from algebra to M-theory. Stephanie Mitchell/Harvard Staff Photographer.

    Mathematicians work to expand their new pictorial mathematical language into other areas.

    A picture is worth 1,000 words, the saying goes, but a group of Harvard-based scientists is hoping that it may also be worth the same number of equations.

    Pictorial laws appear to unify ideas from disparate, interdisciplinary fields of knowledge, linking them beautifully like elements of a da Vinci painting. The group is working to expand the pictorial mathematical language first outlined last year by Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, and postdoctoral fellow Zhengwei Liu.

    “There is one word you can take away from this: excitement,” Jaffe said. “And that’s because we’re not trying just to solve a problem here or there, but we are trying to develop a new way to think about mathematics, through developing and using different mathematical languages based on pictures in two, three, and more dimensions.”

    Last year they created a 3-D language called quon, which they used to understand concepts related to quantum information theory. Now, new research has offered tantalizing hints that quon could offer insights into a host of other areas in mathematics, from algebra to Fourier analysis, as well as in theoretical physics, from statistical physics to string theory. The researchers describe their vision of the project in a paper that appeared Jan. 2 in the journal Proceedings of the National Academy of Sciences.

    “There has been a great deal of evolution in this work over the past year, and we think this is the tip of the iceberg,” Jaffe said. “We’ve discovered that the ideas we used for quantum information are relevant to a much broader spectrum of subjects. We are very grateful to have received a grant from the Templeton Religion Trust that enabled us to assemble a team of researchers last summer to pursue this project further, including undergraduates, graduate students, and postdocs, as well as senior collaborators at other institutions.”

    The core team involves distinguished mathematicians such as Adrian Ocneanu, a visiting professor this year at Harvard, Vaughan Jones, and Alina Vdovina. As important are rising stars who have come to Harvard from around the world, including Jinsong Wu from the Harbin Institute of Technology and William Norledge, a recent graduate from the University of Newcastle. Also involved are students such as Alex Wozniakowski, one of the original members of the project and now a student at Nanyang Technological University in Singapore, visiting graduate students Kaifeng Bu from Zhejiang University in Hangzhou, China, Weichen Gu and Boqing Xue from the Chinese Academy of Sciences in Beijing, Harvard graduate student Sruthi Narayanan, and Chase Bendarz, an undergraduate at Northwestern University and Harvard.

    An illustration of the project is pictured in Lyman Building at Harvard University. Stephanie Mitchell/Harvard Staff Photographer.

    While images have been used in mathematics since ancient times, Jaffe and colleagues believe that the team’s approach, which involves applying pictures to math generally and using images to explore the connections between math and subjects such as physics and cognitive science, may mark the emergence of a new field.

    Among the sort of problems the team has already been able to solve, Liu said, is a pictorial way to think about Fourier analysis.

    “We developed this, motivated by several ideas from Ocneanu,” he said. “Immediately, we used this to give new insights into quantum information. But we also found that we could prove an elaborate algebraic identity for formula 6j-symbols,” a standard tool in representation theory, in theoretical physics, and in chemistry.

    That identity had been found in an elementary case, but Harvard mathematician Shamil Shakirov conjectured that it was true in a general form. The group has now posted a proof on arXiv.org that is under review for publication later in the year. Another very general family of identities that the group has understood simply using the geometric Fourier transform is known as the Verlinde fusion formulas.

    “By looking at the mathematical analysis of pictures, we also found some really unexpected new inequalities. They generalize the famous uncertainty principles of [Werner] Heisenberg and of [Lucien] Hardy and become parts of a larger story,” Liu said. “So the mathematics of the picture languages themselves is quite interesting to understand. We then see their implications on other topics.”

    “I am very taken by this project, because before this, I was working on quantum information, but the only way I knew to do that was using linear algebra,” said Bu. “But working with Arthur and Zhengwei, we’ve been able to use this pictorial language to derive new ideas and geometric tools that we can use to develop new quantum protocols. They have already been useful, and we foresee that these ideas could have wide-ranging applications in the future.

    “It’s amazing, I think, that we can use a simple pictorial language to describe very complicated algebra equations,” Bu continued. “I think this is not only a new approach, but a new field for mathematics.”

    Ocneanu interjected, “Ultimately what higher-dimensional picture language does is to translate the structure of space into mathematics in a natural way.”

    Whereas traditional, linear algebra flattens 3-D concepts into a single line of equations, he said, the picture language allows scientists to use 3-D and higher-dimensional spaces to translate the world around them.

    “Space, or more generally space-time, is a kind of computational machine,” said Ocneanu. “We should really translate what space is doing into the kinds of things mathematicians use, so we can read the structure of space.”

    For Norledge, the new mathematical language is striking in the way it builds from a handful of relatively simple concepts into a complex theory.

    “My background is in representation theory; my thesis is in this area of math called geometric group theory,” he said. “So with a background of using pictures and geometric objects, it helps to apply mathematics in this way. We’re still trying to realize this, but if this all goes through and succeeds, you’ve got a very beautiful area of mathematics where you start with just a few axioms, and just from that beginning you can generalize this highly nontrivial theory with this beautiful structure.”

    “We hope that eventually one can implement the ideas we are studying in new theoretical-physics models, as well as in some practical terms,” Jaffe said. “To share in our excitement, take a look at our website.”

    See the full article here .

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  • richardmitnick 2:51 pm on November 10, 2017 Permalink | Reply
    Tags: , Mathematics, , Taco-The Tensor Algebra Compiler   

    From MIT: “Faster big-data analysis” 

    MIT News
    MIT Widget

    MIT News

    October 30, 2017
    Larry Hardesty

    A new MIT computer system speeds computations involving “sparse tensors,” multidimensional data arrays that consist mostly of zeroes. Image: Christine Daniloff, MIT

    System for performing “tensor algebra” offers 100-fold speedups over previous software packages.

    We live in the age of big data, but most of that data is “sparse.” Imagine, for instance, a massive table that mapped all of Amazon’s customers against all of its products, with a “1” for each product a given customer bought and a “0” otherwise. The table would be mostly zeroes.

    With sparse data, analytic algorithms end up doing a lot of addition and multiplication by zero, which is wasted computation. Programmers get around this by writing custom code to avoid zero entries, but that code is complex, and it generally applies only to a narrow range of problems.

    At the Association for Computing Machinery’s Conference on Systems, Programming, Languages and Applications: Software for Humanity (SPLASH), researchers from MIT, the French Alternative Energies and Atomic Energy Commission, and Adobe Research recently presented a new system that automatically produces code optimized for sparse data.

    That code offers a 100-fold speedup over existing, non-optimized software packages. And its performance is comparable to that of meticulously hand-optimized code for specific sparse-data operations, while requiring far less work on the programmer’s part.

    The system is called Taco, for tensor algebra compiler. In computer-science parlance, a data structure like the Amazon table is called a “matrix,” and a tensor is just a higher-dimensional analogue of a matrix. If that Amazon table also mapped customers and products against the customers’ product ratings on the Amazon site and the words used in their product reviews, the result would be a four-dimensional tensor.

    “Sparse representations have been there for more than 60 years,” says Saman Amarasinghe, an MIT professor of electrical engineering and computer science (EECS) and senior author on the new paper. “But nobody knew how to generate code for them automatically. People figured out a few very specific operations — sparse matrix-vector multiply, sparse matrix-vector multiply plus a vector, sparse matrix-matrix multiply, sparse matrix-matrix-matrix multiply. The biggest contribution we make is the ability to generate code for any tensor-algebra expression when the matrices are sparse.”

    Joining Amarasinghe on the paper are first author Fredrik Kjolstad, an MIT graduate student in EECS; Stephen Chou, also a graduate student in EECS; David Lugato of the French Alternative Energies and Atomic Energy Commission; and Shoaib Kamil of Adobe Research.

    Science paper:
    The Tensor Algebra Compiler

    Custom kernels

    In recent years, the mathematical manipulation of tensors — tensor algebra — has become crucial to not only big-data analysis but machine learning, too. And it’s been a staple of scientific research since Einstein’s time.

    Traditionally, to handle tensor algebra, mathematics software has decomposed tensor operations into their constituent parts. So, for instance, if a computation required two tensors to be multiplied and then added to a third, the software would run its standard tensor multiplication routine on the first two tensors, store the result, and then run its standard tensor addition routine.

    In the age of big data, however, this approach is too time-consuming. For efficient operation on massive data sets, Kjolstad explains, every sequence of tensor operations requires its own “kernel,” or computational template.

    “If you do it in one kernel, you can do it all at once, and you can make it go faster, instead of having to put the output in memory and then read it back in so that you can add it to something else,” Kjolstad says. “You can just do it in the same loop.”

    Computer science researchers have developed kernels for some of the tensor operations most common in machine learning and big-data analytics, such as those enumerated by Amarasinghe. But the number of possible kernels is infinite: The kernel for adding together three tensors, for instance, is different from the kernel for adding together four, and the kernel for adding three three-dimensional tensors is different from the kernel for adding three four-dimensional tensors.

    Many tensor operations involve multiplying an entry from one tensor with one from another. If either entry is zero, so is their product, and programs for manipulating large, sparse matrices can waste a huge amount of time adding and multiplying zeroes.

    Hand-optimized code for sparse tensors identifies zero entries and streamlines operations involving them — either carrying forward the nonzero entries in additions or omitting multiplications entirely. This makes tensor manipulations much faster, but it requires the programmer to do a lot more work.

    The code for multiplying two matrices — a simple type of tensor, with only two dimensions, like a table — might, for instance, take 12 lines if the matrix is full (meaning that none of the entries can be omitted). But if the matrix is sparse, the same operation can require 100 lines of code or more, to track omissions and elisions.

    Enter Taco

    Taco adds all that extra code automatically. The programmer simply specifies the size of a tensor, whether it’s full or sparse, and the location of the file from which it should import its values. For any given operation on two tensors, Taco builds a hierarchical map that indicates, first, which paired entries from both tensors are nonzero and, then, which entries from each tensor are paired with zeroes. All pairs of zeroes it simply discards.

    Taco also uses an efficient indexing scheme to store only the nonzero values of sparse tensors. With zero entries included, a publicly released tensor from Amazon, which maps customer ID numbers against purchases and descriptive terms culled from reviews, takes up 107 exabytes of data, or roughly 10 times the estimated storage capacity of all of Google’s servers. But using the Taco compression scheme, it takes up only 13 gigabytes — small enough to fit on a smartphone.

    “Many research groups over the last two decades have attempted to solve the compiler-optimization and code-generation problem for sparse-matrix computations but made little progress,” says Saday Sadayappan, a professor of computer science and engineering at Ohio State University, who was not involved in the research. “The recent developments from Fred and Saman represent a fundamental breakthrough on this long-standing open problem.”

    “Their compiler now enables application developers to specify very complex sparse matrix or tensor computations in a very easy and convenient high-level notation, from which the compiler automatically generates very efficient code,” he continues. “For several sparse computations, the generated code from the compiler has been shown to be comparable or better than painstakingly developed manual implementations. This has the potential to be a real game-changer. It is one of the most exciting advances in recent times in the area of compiler optimization.”

    See the full article here .

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