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  • richardmitnick 4:40 pm on February 18, 2020 Permalink | Reply
    Tags: , , , Mathematics, MIP-multiprover interactive proof, , , ,   

    From Science News: “How a quantum technique highlights math’s mysterious link to physics” 

    From Science News

    February 17, 2020
    Tom Siegfried

    Verifying proofs to very hard math problems is possible with infinite quantum entanglement.

    1
    A technique that relies on quantum entanglement (illustrated) expands the realm of mathematical problems for which the solution could (in theory) be verified. inkoly/iStock/Getty Images Plus.

    It has long been a mystery why pure math can reveal so much about the nature of the physical world.

    Antimatter was discovered in Paul Dirac’s equations before being detected in cosmic rays. Quarks appeared in symbols sketched out on a napkin by Murray Gell-Mann several years before they were confirmed experimentally. Einstein’s equations for gravity suggested the universe was expanding a decade before Edwin Hubble provided the proof. Einstein’s math also predicted gravitational waves a full century before behemoth apparatuses detected those waves (which were produced by collisions of black holes — also first inferred from Einstein’s math).

    Nobel laureate physicist Eugene Wigner alluded to math’s mysterious power as the “unreasonable effectiveness of mathematics in the natural sciences.” Somehow, Wigner said, math devised to explain known phenomena contains clues to phenomena not yet experienced — the math gives more out than was put in. “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and … there is no rational explanation for it,” Wigner wrote in 1960.

    But maybe there’s a new clue to what that explanation might be. Perhaps math’s peculiar power to describe the physical world has something to do with the fact that the physical world also has something to say about mathematics.

    At least that’s a conceivable implication of a new paper that has startled the interrelated worlds of math, computer science and quantum physics.

    In an enormously complicated 165-page paper, computer scientist Zhengfeng Ji and colleagues present a result that penetrates to the heart of deep questions about math, computing and their connection to reality. It’s about a procedure for verifying the solutions to very complex mathematical propositions, even some that are believed to be impossible to solve. In essence, the new finding boils down to demonstrating a vast gulf between infinite and almost infinite, with huge implications for certain high-profile math problems. Seeing into that gulf, it turns out, requires the mysterious power of quantum physics.

    Everybody involved has long known that some math problems are too hard to solve (at least without unlimited time), but a proposed solution could be rather easily verified. Suppose someone claims to have the answer to such a very hard problem. Their proof is much too long to check line by line. Can you verify the answer merely by asking that person (the “prover”) some questions? Sometimes, yes. But for very complicated proofs, probably not. If there are two provers, though, both in possession of the proof, asking each of them some questions might allow you to verify that the proof is correct (at least with very high probability). There’s a catch, though — the provers must be kept separate, so they can’t communicate and therefore collude on how to answer your questions. (This approach is called MIP, for multiprover interactive proof.)

    Verifying a proof without actually seeing it is not that strange a concept. Many examples exist for how a prover can convince you that they know the answer to a problem without actually telling you the answer. A standard method for coding secret messages, for example, relies on using a very large number (perhaps hundreds of digits long) to encode the message. It can be decoded only by someone who knows the prime factors that, when multiplied together, produce the very large number. It’s impossible to figure out those prime numbers (within the lifetime of the universe) even with an army of supercomputers. So if someone can decode your message, they’ve proved to you that they know the primes, without needing to tell you what they are.

    Someday, though, calculating those primes might be feasible, with a future-generation quantum computer. Today’s quantum computers are relatively rudimentary, but in principle, an advanced model could crack codes by calculating the prime factors for enormously big numbers.

    That power stems, at least in part, from the weird phenomenon known as quantum entanglement. And it turns out that, similarly, quantum entanglement boosts the power of MIP provers. By sharing an infinite amount of quantum entanglement, MIP provers can verify vastly more complicated proofs than nonquantum MIP provers.

    It is obligatory to say that entanglement is what Einstein called “spooky action at a distance.” But it’s not action at a distance, and it just seems spooky. Quantum particles (say photons, particles of light) from a common origin (say, both spit out by a single atom) share a quantum connection that links the results of certain measurements made on the particles even if they are far apart. It may be mysterious, but it’s not magic. It’s physics.

    Say two provers share a supply of entangled photon pairs. They can convince a verifier that they have a valid proof for some problems. But for a large category of extremely complicated problems, this method works only if the supply of such entangled particles is infinite. A large amount of entanglement is not enough. It has to be literally unlimited. A huge but finite amount of entanglement can’t even approximate the power of an infinite amount of entanglement.

    As Emily Conover explains in her report for Science News, this discovery proves false a couple of widely believed mathematical conjectures. One, known as Tsirelson’s problem, specifically suggested that a sufficient amount of entanglement could approximate what you could do with an infinite amount. Tsirelson’s problem was mathematically equivalent to another open problem, known as Connes’ embedding conjecture, which has to do with the algebra of operators, the kinds of mathematical expressions that are used in quantum mechanics to represent quantities that can be observed.

    Refuting the Connes conjecture, and showing that MIP plus entanglement could be used to verify immensely complicated proofs, stunned many in the mathematical community. (One expert, upon hearing the news, compared his feces to bricks.) But the new work isn’t likely to make any immediate impact in the everyday world. For one thing, all-knowing provers do not exist, and if they did they would probably have to be future super-AI quantum computers with unlimited computing capability (not to mention an unfathomable supply of energy). Nobody knows how to do that in even Star Trek’s century.

    Still, pursuit of this discovery quite possibly will turn up deeper implications for math, computer science and quantum physics.

    It probably won’t shed any light on controversies over the best way to interpret quantum mechanics, as computer science theorist Scott Aaronson notes in his blog about the new finding. But perhaps it could provide some sort of clues regarding the nature of infinity. That might be good for something, perhaps illuminating whether infinity plays a meaningful role in reality or is a mere mathematical idealization.

    On another level, the new work raises an interesting point about the relationship between math and the physical world. The existence of quantum entanglement, a (surprising) physical phenomenon, somehow allows mathematicians to solve problems that seem to be strictly mathematical. Wondering why physics helps out math might be just as entertaining as contemplating math’s unreasonable effectiveness in helping out physics. Maybe even one will someday explain the other.

    See the full article here .


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  • richardmitnick 12:14 pm on February 12, 2020 Permalink | Reply
    Tags: Atom or noise?, , , , Mathematics, , , Stanford’s Department of Bioengineering   

    From SLAC National Accelerator Lab: “Atom or noise? New method helps cryo-EM researchers tell the difference” 

    From SLAC National Accelerator Lab

    February 11, 2020
    Nathan Collins

    Cryogenic electron microscopy can in principle make out individual atoms in a molecule, but distinguishing the crisp from the blurry parts of an image can be a challenge. A new mathematical method may help.

    Cryogenic electron microscopy, or cryo-EM, has reached the point where researchers could in principle image individual atoms in a 3D reconstruction of a molecule – but just because they could see those details doesn’t always mean they do. Now, researchers at the Department of Energy’s SLAC National Accelerator Laboratory and Stanford University have proposed a new way to quantify how accurate such reconstructions are and, in the process, how confident they can be in their molecular interpretations. The study was published February 10 in Nature Methods.

    Cryo-EM works by freezing biological molecules which can contain thousands of atoms so they can be imaged under an electron microscope. By aligning and combining many two-dimensional images, researchers can compute three-dimensional maps of an entire molecule, and this technique has been used to study everything from battery failure to the way viruses invade cells. However, an issue that has been hard to solve is how to accurately assess the true level of detail or resolution at every point in such maps and in turn determine what atomic features are truly visible or not.

    1
    A cryo-EM map of the molecule apoferritin (left) and a detail of the map showing the atomic model researchers use to construct Q-scores. (Image courtesy Greg Pintilie)

    Wah Chiu, a professor at SLAC and Stanford, Grigore Pintilie, a computational scientist in Chiu’s group, and colleagues devised the new measures, known as Q-scores, to address that issue. To compute Q-scores, scientists start by building and adjusting an atomic model until it best matches the corresponding cryo-EM derived 3D map. Then, they compare the map to an idealized version in which each atom is well-resolved, revealing to what degree the map truly resolves the atoms in the atomic model.

    The researchers validated their approach on large molecules, including a protein called apoferritin that they studied in the Stanford-SLAC Cryo-EM Facilities. Kaiming Zhang, another research scientist in Chiu’s group, produced 3D maps close to the highest resolution reached to date – up to 1.75 angstrom, less than a fifth of a nanometer. Using such maps, they showed how Q-scores varied in predictable ways based on overall resolution and on which parts of a molecule they were studying. Pintilie and Chiu say they hope Q-scores will help biologists and others using cryo-EM better understand and interpret the 3D maps and resulting atomic models.

    The study was performed in collaboration with researchers from Stanford’s Department of Bioengineering. Molecular graphics and analysis were performed using the University of California, San Francisco’s Chimera software package. The project was funded by the National Institutes of Health.

    See the full article here .


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    SLAC/LCLS


    SLAC/LCLS II projected view


    SLAC is a multi-program laboratory exploring frontier questions in photon science, astrophysics, particle physics and accelerator research. Located in Menlo Park, California, SLAC is operated by Stanford University for the DOE’s Office of Science.

    SSRL and LCLS are DOE Office of Science user facilities.

     
  • richardmitnick 12:57 pm on February 10, 2020 Permalink | Reply
    Tags: "Where math meets physics", , , Mathematics, ,   

    From Penn Today: “Where math meets physics” 


    From Penn Today

    February 7, 2020
    Erica K. Brockmeier
    Eric Sucar, Photographer

    Collaborations between physicists and mathematicians at Penn showcase the importance of research that crosses the traditional boundaries that separate fields of science.

    1
    Penn is home to an active and flourishing collaboration between physicists and mathematicians. Advances in the fields of geometry, string theory, and particle physics have been made possible by teams of researchers, like physicist Burt Ovrut (above), who speak different “languages,” embrace new research cultures, and understand the power of tackling problems through an interdisciplinary approach.

    In the scientific community, “interdisciplinary” can feel like an overused, modern-day buzzword. But uniting different academic disciplines is far from a new concept. Math, chemistry, physics, and biology were grouped together for many years under the umbrella “natural philosophy,” and it was only as knowledge grew and specialization became necessary that these disciplines became more specialized.

    With many complex scientific questions still in need of answers, working across multiple fields is now seen an essential part of research. At Penn, long-running collaborations between the physics and astronomy and the math departments showcase the importance of interdisciplinary research that crosses traditional boundaries. Advances in geometry, string theory, and particle physics, for example, have been made possible by teams of researchers who speak different “languages,” embrace new research cultures, and understand the power of tackling problems through an interdisciplinary approach.

    A tale of two disciplines

    Math and physics are two closely connected fields. For physicists, math is a tool used to answer questions. For example, Newton invented calculus to help describe motion. For mathematicians, physics can be a source of inspiration, with theoretical concepts such as general relativity and quantum theory providing an impetus for mathematicians to develop new tools.

    But despite their close connections, physics and math research relies on distinct methods. As the systematic study of how matter behaves, physics encompasses the study of both the great and the small, from galaxies and planets to atoms and particles. Questions are addressed using combinations of theories, experiments, models, and observations to either support or refute new ideas about the nature of the universe.

    In contrast, math is focused on abstract topics such as quantity (number theory), structure (algebra), and space (geometry). Mathematicians look for patterns and develop new ideas and theories using pure logic and mathematical reasoning. Instead of experiments or observations, mathematicians use proofs to support their ideas.

    While physicists rely heavily on math for calculations in their work, they don’t work towards a fundamental understanding of abstract mathematical ideas in the way that mathematicians do. Physicists “want answers, and the way they get answers is by doing computations,” says mathematician Tony Pantev. “But in mathematics, the computations are just a decoration on top of the cake. You have to understand everything completely, then you do a computation.”

    This fundamental difference leads researchers in both fields to use the analogy of language, highlighting a need to “translate” ideas in order to make progress and understand one another. “We are dealing with how to formulate physics questions so it can be seen as a mathematics problem” says physicist Mirjam Cvetič . “That’s typically the hardest part.”

    3
    Kamien works on physics problems in that have a strong connection to geometry and topology and encourages his students to understand problems as mathematicians do. “Understanding things for the sake of understanding them is worthwhile, and connecting them to things that other people know is also worthwhile,” he says.

    “A physicist comes to us, asks, ‘How do you prove that this is true?’ and we immediately show them it’s false,” says mathematician Ron Donagi. “But we keep talking, and the trick is not to do what they say to do but what they mean, a translation of the problem.”

    In addition to differences in methodology and language, math and physics also have different research cultures. In physics, papers might involve dozens of co-authors and institutions, with researchers publishing work several times per year. In contrast, mathematicians might work on a single problem that takes years to complete with a small number of collaborators. “Sometimes, physics papers are essentially, ‘We discovered this thing, isn’t that cool,’” says physicist Randy Kamien. “But math is never like that. Everything is about understanding things for the sake of understanding them. Culturally, it’s very different.”

    Mind the gap

    When asked how mathematicians and physicists can bridge these fundamental gaps and successfully work together, many researchers refer to a commonly cited example that also has a connection to Penn. In the 1950s, Eugenio Calabi, now professor emeritus, conjectured the existence of a six-dimensional manifold, a topological space arranged in a way that allows complex structures to be described and understood more simply. After the manifold’s existence was proven in 1978 by Shing-Tung Yau, this new finding was poised to become a fundamental component of a new idea in particle physics: string theory.

    Proposed in the 1970s as a candidate framework for a “theory of everything,” it describes matter as being made of one-dimensional vibrating strings that form elementary particles, like electrons and neutrinos, as well as forces, like gravity and electromagnetism. The challenge, however, is that string theory requires a 10-dimensional universe, so physicists turned to the Calabi-Yau manifolds as a place to house the “extra” dimensions.

    Because the structure is so complex and only recently proven by mathematicians, it wasn’t simple to directly implement into a physics framework, even though physicists use math all of the time in their work. Physicists “use differential geometry, but that’s been known for a long time,” says physicist Burt Ovrut. “When all of a sudden string theory launches, who the heck knows what a Calabi-Yau manifold is?”

    Through the combined efforts of Ed Witten, a physicist with strong mathematical knowledge, and mathematician Michael Atiyah, researchers found a way to apply Calabi-Yau manifolds in string theory. It was the ability of Witten to help translate ideas between the two fields that many researchers say was instrumental in successfully applying brand-new ideas from mathematics into up-and-coming theories from physics.

    At Penn, mathematicians, including Donagi, Pantev, and Antonella Grassi, and physicists Cvetič , Kamien, Ovrut, and Jonathan Heckman have also recognized the importance of speaking a common language as they work across the two fields. They credit Penn as being a place that’s particularly adept at fostering connections and bridging gaps in cultural, linguistic, and methodological differences, and they credit their success to time spent listening to new ideas and developing ways to “translate” between languages.

    For Donagi, it was a chance encounter with Witten in the mid 1990s that led the mathematician to his first collaboration with a researcher outside of pure math. He enjoyed working with Witten so much that he reached out to Penn physicists Cvetič and Ovrut to start a “local” crossover collaboration. “I’ve been hooked since then, and I’ve been talking as much to physicists as to other mathematicians,” Donagi says.

    During the mid-2000s, Donagi and Ovrut co-led a math and physics program with Pantev and Grassi that was supported by the U.S. Department of Energy. The collaboration marked a successful first official math and physics crossover collaboration at Penn. As Ovrut explains, the work was focused on a specific kind of string theory and required extremely close interactions between physics and math researchers. “It was at the very edge of mathematics and algebraic geometry, so I couldn’t do this myself, and the mathematicians were very interested in these things.”

    Cvetič, a longtime collaborator with Donagi and Grassi, says that Penn’s mathematicians have the expertise they need to help answer important questions in physics and that their collaborations at the interface of string theory and algebraic geometry are “extremely fruitful and productive.”

    “I think it’s been incredibly productive and helpful for both our groups,” Donagi says. “We’ve been doing this for longer than anyone else, and we have a really good strong connection between the groups. They’ve almost become one group.”

    3
    “What facilitates this type of research is that we can talk to the physicists,” says Pantev (right), who has worked for many years with Cvetič and Donagi. “When we go talk to them, they know how to speak our language, and they can explain the questions they are struggling with in a way that we can understand and approach them.”

    And in terms of embracing cultural differences, physicists like Kamien, who works on problems with a strong connection to geometry and topology, encourages his group members to try to understand math the way mathematicians do instead of only seeing it as a tool for their work. “We’ve tried to absorb not just their language but their culture, how they understand things, how sometimes understanding a problem more deeply is better,” he says.

    Crossing paths

    Craig Lawrie and Ling Lin, a current and former postdoc working with Cvetič and Heckman, know firsthand about both the challenges and opportunities of working on a problem that combines both cutting-edge math and physics. Physicists like Lawrie and Lin, who work in M-theory and F-theory, are trying to figure out what types of particles different geometric structures can create while also removing the “extra” six dimensions.

    Adding extra symmetries makes string theory problems easier to work with and allows researchers to ask questions about the properties of geometric structures and how they correspond to real-world physics. Building off previous work by Heckman, Lawrie and Lin were able to extract physical features from known geometries in five-dimensional systems to see if those particles overlapped with standard model particles. Using their knowledge of both physics and math, the researchers showed that geometries in different dimensions are all related mathematically, which means they can study particles in different dimensions more easily.

    Using their physics intuition, Lawrie and Lin were able to apply their knowledge of math to make new discoveries that wouldn’t have been possible if the two fields were used in isolation. “What we found seems to suggest that theories in five dimensions come from theories in six dimensions,” explains Lin. “That is something that mathematicians, if they didn’t know about string theory or physics, would not think about.”

    Lawrie adds that being able to work directly with mathematicians is also helpful in their field since understanding new math research can be a challenge, even for theoretical physics researchers. “As physicists, we can have a long discussion where we use a lot of intuition, but if you talk to a mathematician they will say, ‘Wait, precisely what do you mean by that?’ and then you have to pull out your important assumptions,” says Lawrie. “It’s also good for clarifying our own thought process.”

    Rodrigo Barbosa also knows what it’s like to work across fields, in his case coming from math to physics. While studying a seven-dimensional manifold as part of his Ph.D. program, Barbosa connected at a conference with Lawrie over their shared research interests. They were then able to combine their experiences through a successful interdisciplinary collaboration [Physical Review D], work that was motivated by Barbosa’s Ph.D. research in math that included both junior and senior faculty as well as postdocs and graduate students from physics.

    While Barbosa says that the work was challenging, especially being the only mathematician in the group, he also found it rewarding. He enjoyed being able to provide mathematical explanations for certain difficult concepts and relished the rare opportunity to work so closely with researchers outside of his field while still in graduate school. “I’m very grateful that I did my Ph.D. at Penn because it’s really one of a handful of places where this could have happened,” he says.

    The next generation

    Faculty in both departments see the next generation of students and postdocs as “ambidextrous,” having fundamental skills, knowledge, and intuition from both math and physics. “Young people are extremely sophisticated and open minded,” says Pantev. “In the old days, it was very hard to get into physics-related research if you were a mathematician because the thinking is completely different. Now, young people are equally versed in both modes of thinking, so it’s easy for them to make progress.”

    4
    Heckman joined the physics faculty in 2017 and is already active in a number of collaborations with the math department. “What makes this place so great is that we’re talking a common language,” he says. “Although Ron says we sometimes speak with an accent.”

    Heckman is also a member of this new ambidextrous generation of researchers, and in his two years at Penn he has co-authored several papers and started new projects with mathematicians. He says that researchers who want to be successful in the future need to be able to balance the needs of both fields. “Some students act more like mathematicians, and I have to guide them to act more like physicists, and others have more physical intuition but they have to pick up the math,” he says.

    It’s a balance that requires a blend of flexibility and precision, and is one that will be a continuing challenge as topics become increasingly complex and new observations are made from physics experiments. “Mathematicians want to make everything well-defined and rigorous. From a physics perspective, sometimes you want to get an answer that doesn’t need to be well-defined, so you need to make a compromise,” says Lin.

    This compromise is something that’s attracted Barbosa to working more with physicists, adding that the two fields are complementary. “Problems have become so difficult that you need input from all possible directions. Physics works by finding examples and describing solutions, while in math you try to see how general these equations are and how things fit together,” Barbosa says. He also enjoys that physics provides him with a way to make progress on answering questions more quickly than in pure math, where problems can take years to solve.

    The future of crossing over

    The future of interdisciplinary research will depend a lot on the next generation, but Penn is well positioned to continue leading these efforts thanks to the proximity of the two departments, shared grants, cross-listed courses, and students and postdocs that actively work on problems across fields. “There is this constant osmosis of basic knowledge that builds up students who are literate and comfortable with sophisticated language,” says Pantev. “I think we are ahead of the curve, and I think we’ll stay ahead of the curve.”

    It’s something that many at Penn agree is a unique feature of their two departments. “It’s very rare to have such close relationships between mathematicians who really listen to what we say,” says Ovrut. “Penn should be proud of itself for having that kind of synergy. It is not something you see every day.”

    6
    Ovrut (left) was one of the co-leads, along with Donagi, of the incredibly successful joint math and physics program, the first official collaboration between the two departments at Penn.

    See the full article here .

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    U Penn campus

    Academic life at Penn is unparalleled, with 100 countries and every U.S. state represented in one of the Ivy League’s most diverse student bodies. Consistently ranked among the top 10 universities in the country, Penn enrolls 10,000 undergraduate students and welcomes an additional 10,000 students to our world-renowned graduate and professional schools.

    Penn’s award-winning educators and scholars encourage students to pursue inquiry and discovery, follow their passions, and address the world’s most challenging problems through an interdisciplinary approach.

     
  • richardmitnick 11:46 am on February 9, 2020 Permalink | Reply
    Tags: Chenyang Xu applies the techniques of abstract algebra to study concrete but complex geometric objects., Mathematics,   

    From MIT News: “New theories at the intersection of algebra and geometry” 

    MIT News

    From MIT News

    February 8, 2020
    Jonathan Mingle

    1
    Chenyang Xu. Image: M. Scott Brauer

    Professor Chenyang Xu applies the techniques of abstract algebra to study concrete but complex geometric objects.

    As a self-described “classical type of mathematician,” Chenyang Xu eschews software for paper and pen, chalk and chalkboard. Walk by his office, and you might simply see him pacing about, deep in concentration.

    Walking — across campus to get a cup of coffee, or from his apartment to his office — is an essential part of his process.

    “The way I think about math, I do a lot of picturing in my brain,” he says. “If I need a more clear picture, I might draw something and do some calculations. And when I walk I think of these pictures.”

    Those paces sometimes lead him to colleagues’ offices. “There are so many great minds here, and I interact with my colleagues in the department a lot,” says Xu, a recently tenured professor of mathematics at MIT.

    Xu’s specialty is algebraic geometry, which applies the problem-solving methods of abstract algebra to the complex but concrete shapes, surfaces, spaces, and curves of geometry. His primary objects of study are algebraic varieties — geometric manifestations of sets of solutions of systems of polynomial equations. As he walks and talks with colleagues, Xu focuses on ways of classifying these algebraic varieties in higher dimensions, using the techniques of birational geometry.

    “I like to talk with other mathematicians working in my subject,” Xu says. “We discuss a bit, then go back to think for ourselves, encounter new difficulties, then discuss again. Most of my papers are basically collaborations.”

    Such a collaboration helped Xu take his research in a new direction toward developing the new theory of K-stability of Fano varieties. Eight years ago, he devoted some thought to a certain subject in his field known as K-stability, which he describes as “an algebraic definition invented for differential geometry studies.”

    “I tried to develop an algebraic theory based on this K-stability as a background intuition, using algebraic geometry tools.” After a few years’ “gap,” he eventually came back to it because of conversations with his collaborator Chi Li, a professor of mathematics at Purdue University.

    “He had more of a differential geometry background and translated that concept into algebraic geometry,” says Xu. “That’s when I realized this was important to study. Since then, we have done more than we expected four or five years ago.”

    Together they published a highly cited paper Annals of Mathematics in 2014 on the “K-stability of Fano varieties,” which put forward an entirely new theory in the field of birational algebraic geometry.

    It was representative of his approach to mathematics, which involves advancing new theories before tackling specific problems.

    “In my subject there are questions that everybody trying to solve, that have been open for 40 years,” Xu says. “I have those kinds of problems in my mind. My way of doing math is to go after the theory. Instead of working on one problem with techniques, we have to first develop the theory. We then see something in a new light. Every time I find some new theory, I test it on old classical problems to see if it works or not.”

    The beauty of math

    Growing up near Chengdu, in China’s Sichuan Province, Xu enjoyed math from a young age. “I attended some math Olympiads, and I did okay, but I wasn’t the gold medal winner,” he says with a laugh.

    He was talented enough, however, to earn bachelor’s and master’s degrees at Peking University, as a part of the premier math program in China.

    “After I got into college, I started to learn more advanced mathematics, and I found it very beautiful and very deep,” he says. “To me, a big chunk of mathematics is art more than science.”

    Toward the end of his time at Peking, he concentrated increasingly on algebraic geometry. “I just like geometry a lot and wanted to study some subject related to geometry,” he says. “I found that I’m good at the techniques of algebra. So using those techniques to study geometry fit me very well.”

    Xu then pursued a PhD at Princeton University, where his advisor, János Kollár, a leading algebraic geometer, had a “huge influence” on him.

    “What I learned from him, aside from many techniques, of course, was more about what I could call ‘taste,’” says Xu. “What questions are important in mathematics? In general, graduate students or postdocs in the early stages of their career need some role model to follow. Doing math is a complicated thing, and at some point there are choices they need to make,” he says, that require balancing how difficult or interesting a particular problem might be with more practical concerns about its tractability.

    In addition to Kollár’s mentorship, the unfamiliarity of his new surroundings also aided his research.

    “I had never been outside China before that point, so there was a bit of culture shock,” he recalls. “I didn’t know much about U.S. culture at the time. But in some sense that made me even more concentrated on my work.”

    After Xu received his doctorate in 2008, he spent three years as a postdoc and C.L.E. Moore Instructor at MIT. He then spent about six years as a professor at the Beijing International Center of Mathematical Research and then returned to MIT as a full professor of mathematics in 2018.

    Throughout those years, Xu demonstrated a talent for finding important questions to pursue, becoming a leading thinker in his field and making a series of major advances in algebraic birational geometry.

    In 2017, Xu won the inaugural Future Science Prize in Mathematics and Computer Science for his “fundamental contributions” to the field of birational geometry. Some of that field’s real-world applications include coding and robotics. For example, birational geometry techniques are used to help robots “see” by grouping a series of two-dimensional pictures together into something approximating a field of vision to navigate our three-dimensional world.

    Xu’s work to advance the minimal model program (MMP) — a key theory in birational geometry that was first articulated in the early 1980s — and apply it to algebraic varieties won him the 2019 New Horizons Prize for early-career achievement in mathematics. He has since proved a series of conjectures related to the MMP, expanding it to previously untested varieties of certain conditions.

    The theory of algebraic K-stability that he developed has proven to be fertile ground for new discoveries. “I’m still working on this topic, and it’s a particularly interesting question to me,” he says.

    Xu has been making progress on proving other key conjectures related to K-stability rooted in the minimal model program. Recently, he drew on that prior work to prove the existence of moduli space for Fano algebraic varieties. Now he’s hard at work developing a solution for a specific property of that moduli space: its “compactness.”

    “To solve that problem it will be very important,” he says. “I hope we can still solve the last piece of it. I’m pretty sure that would be my best work to date.”

    See the full article here .


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  • richardmitnick 1:09 pm on January 22, 2020 Permalink | Reply
    Tags: "Brewing a better espresso with a shot of maths", , Mathematics,   

    From University of Portsmouth: “Brewing a better espresso with a shot of maths” 

    From From University of Portsmouth

    22 January 2020

    Read how Dr Jamie Foster’s number-crunching has uncovered the secret to espresso perfection.

    1
    Dr Jamie Foster

    Mathematicians, physicists and materials experts might not spring to mind as the first people to consult about whether you are brewing your coffee right.

    But a team of such researchers including Dr Jamie Foster, a mathematician at the University of Portsmouth’s School of Mathematics and Physics, are challenging common espresso wisdom.

    They have found, that fewer coffee beans, ground more coarsely, are the key to a drink that is cheaper to make, more consistent from shot to shot, and just as strong.

    The study is published in the journal Matter.

    Dr Foster and colleagues set out wanting to understand why sometimes two shots of espresso, made in seemingly the same way, can sometimes taste rather different.

    2
    Researchers have found that fewer coffee beans, ground more coarsely, are the key to a drink that is cheaper to make, more consistent from shot to shot, and just as strong.

    They began by creating a new mathematical theory to describe extraction from a single grain, many millions of which comprise a coffee ‘bed’ which you would find in the basket of an espresso machine.

    Dr Foster said: “In order to solve the equations on a realistic coffee bed you would need an army of super computers, so we needed to find a way of simplifying the equations.

    “The hard mathematical work was in making these simplifications systematically, in such a way that none of the important detail was lost.

    “The conventional wisdom is that if you want a stronger cup of coffee, you should grind your coffee finer. This makes sense because the finer the grounds mean that more surface area of coffee bean is exposed to water, which should mean a stronger coffee.”

    When the researchers began to look at this in detail, it turned out to be not so simple. They found coffee was more reliable from cup to cup when using fewer beans ground coarsely.

    “When beans were ground finely, the particles were so small that in some regions of the bed they clogged up the space where the water should be flowing,” Dr Foster said.

    “These clogged sections of the bed are wasted because the water cannot flow through them and access that tasty coffee that you want in your cup. If we grind a bit coarser, we can access the whole bed and have a more efficient extraction.

    “It’s also cheaper, because when the grind setting is changed, we can use fewer beans and be kinder to the environment.

    “Once we found a way to make shots efficiently, we realised that as well as making coffee shots that stayed reliably the same, we were using less coffee.”

    The new recipes have been trialled in a small US coffee shop over a period of one year and they have reported saving thousands of dollars. Estimates indicates that scaling this up to encompass the whole US coffee market could save over $US1.1bn dollar per year.

    Previous studies have looked at drip filter coffee. This is the first time mathematicians have used theoretical modelling to study the science of the perfect espresso – a more complicated process due to the additional pressure.

    See the full article here .

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    The University of Portsmouth is a public university in the city of Portsmouth, Hampshire, England. The history of the university dates back to 1908, when the Park building opened as a Municipal college and public library. It was previously known as Portsmouth Polytechnic until 1992, when it was granted university status through the Further and Higher Education Act 1992. It is ranked among the Top 100 universities under 50 in the world.

    We’re a New Breed of University
    We’re proud to be a breath of fresh air in the academic world – a place where everyone gets the support they need to achieve their best.
    We’re always discovering. Through the work we do, we engage with our community and world beyond our hometown. We don’t fit the mould, we break it.
    We educate and transform the lives of our students and the people around us. We recruit students for their promise and potential and for where they want to go.
    We stand out, not just in the UK but in the world, in innovation and research, with excellence in areas from cosmology and forensics to cyber security, epigenetics and brain tumour research.
    Just as the world keeps moving, so do we. We’re closely involved with our local community and we take our ideas out into the global marketplace. We partner with business, industry and government to help improve, navigate and set the course for a better future.
    Since the first day we opened our doors, our story has been about looking forward. We’re interested in the future, and here to help you shape it.

    The university offers a range of disciplines, from Pharmacy, International relations and politics, to Mechanical Engineering, Paleontology, Criminology, Criminal Justice, among others. The Guardian University Guide 2018 ranked its Sports Science number one in England, while Criminology, English, Social Work, Graphic Design and Fashion and Textiles courses are all in the top 10 across all universities in the UK. Furthermore, 89% of its research conducted in Physics, and 90% of its research in Allied Health Professions (e.g. Dentistry, Nursing and Pharmacy) have been rated as world-leading or internationally excellent in the most recent Research Excellence Framework (REF2014).

    The University is a member of the University Alliance and The Channel Islands Universities Consortium. Alumni include Tim Peake, Grayson Perry, Simon Armitage and Ben Fogle.

    Portsmouth was named the UK’s most affordable city for students in the Natwest Student Living Index 2016. On Friday 4 May 2018, the University of Portsmouth was revealed as the main shirt sponsor of Portsmouth F.C. for the 2018–19, 2019–20 and 2020–21 seasons.

     
  • 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

    1
    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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    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 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

    1
    MIT mathematician Andrew “Drew” Sutherland solved a 65-year-old problem about 42. Image: Department of Mathematics

    2
    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

    1
    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”.

    5
    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

    2
    LAGEOS satellite, courtesy of NASA

    3
    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

    1
    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 .


    five-ways-keep-your-child-safe-school-shootings
    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.

    MIT Campus

     
  • 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


    CMNS

    April 16, 2019

    Abby Robinson
    301-405-5845
    abbyr@umd.edu

    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.

    1
    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.”

    3
    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.

    4
    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 .

    five-ways-keep-your-child-safe-school-shootings

    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.

     
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