On Cracking a 50-Year-Old Math Problem, by Claudia Clark
In this article, writer Erica Klarreich provides a primer on the Kadison-Singer problem—an important 50-year-old problem in C*-algebras—and explains how it came to be solved in 2013 by three "computer scientists who had barely a nodding acquaintance with the disciplines at the heart of the problem." The question that Richard Kadison and Isadore Singer "posed in 1959 asks how much it is possible to learn about a 'state' of a quantum system if you have complete information about that state in a special subsystem." Klarreich writes,
more mathematically inclined physicists pounced on the Kadison-Singer problem, which they understood as a question about C*-algebras, abstract structures that capture the algebraic properties not just of quantum systems but also of the random variables used in probability theory, the blocks of numbers called matrices, and regular numbers... Then in 1979, Joel Anderson, now an emeritus professor at Pennsylvania State University, popularized the problem by proving that it is equivalent to an easily stated question about when matrices can be broken down into simpler chunks... Suddenly, the Kadison-Singer problem was everywhere. Over the decades that followed, it emerged as the key problem in one field after another.
Eventually computer scientist Daniel Spielman discovered an equivalent formulation of the problem, devised by Nik Weaver, that was "nearly identical to a simple question about networks:... If there are enough ways to get around in a network—if no individual edge is too important—can the network be broken down into two subnetworks with similar electrical properties?" Five years later, Spielman, working with Adam Marcus and Nikhil Srivastava, proved that the answer to this question is yes, solving the Kadison-Singer problem and its many equivalent formulations. (Photo: Computer scientists (from left), Nikhil Srivastava, Adam Marcus, and Daniel Spielman. Photo by Nikhil Srivastava.)
See "'Outsiders' Crack 50-Year-Old Math Problem," by Erica Klarreich, Quanta Magazine, 24 November 2015.
--- Claudia Clark
On Lewis Caroll, by Lisa DeKeukelaere
On the 150th anniversary of the publication of Alice's Adventures in Wonderland, the journal Nature provides an overview of author Lewis Carroll's contributions to mathematics. Carroll published multiple books and articles under his real name, Charles Dodgson, while he was a mathematics lecturer at the University of Oxford. In addition to subtle mathematical references in Alice's Adventures, Carroll wrote a book of "droll mathematical stories" and puzzles called A Tangled Tale, both of which dabble in wordplay and refer to problems as knots. He believed that such puzzles were useful not only for entertainment, but also for increasing the solver's sense of power. On the academic side, Carroll's publication record includes contributions in rapid arithmetic processes, such as for computing the day of the week for future dates, as well as in using logic trees and creating voting methods.
See "Mathematics: Logic and Lewis Carroll," by Francine F. Abeles. Nature, 19 November 2015, pages 302-304.
--- Lisa DeKeukeleare
Working through Einstein's first proof, by Mike Breen
November marked the 100th anniversary of Albert Einstein's proof of the theory of general relativity. In this article Steve Strogatz gives a wonderful account of Einstein's first proof--a proof of the Pythagorean Theorem, which most of would find to be at least a bit more accessible than that of general relativity. Strogatz walks us through the elegant proof that uses similar right triangles and symmetry. Given a right triangle, the two right triangles that result from drawing the altitude from the original triangle's right angle are similar to one another and similar to the original right triangle. Strogatz writes, "Because the triangles are similar, each occupies the same fraction f of the area of the square on its hypotenuse." Thus the areas are fa^{2}, fb^{2}, and fc^{2}. The two smaller triangles comprise the larger, so fa^{2} + fb^{2} = fc^{2}, and dividing by f gives the result. In the last part of the article, Strogatz notes the importance of symmetry in Einstein's later work and the admitted difficulty that Einstein had with higher mathematics. See "Einstein's First Proof," by Steven Strogatz. The New Yorker, 19 November 2015. For more on Einstein and math, see "How Einstein inspired me to find a magical piece of mathematics," by S. James Gates, Jr. PBS NewsHour, 24 November 2015. |
--- Mike Breen (posted 12/1/15)
On Katherine Johnson, Presidential Medal of Freedom Recipient, by Annette Emerson
Katherine Johnson: America's First Space Flight
Katherine Johnson, longtime mathematician at NASA's Langley Research Center, is a recipient of the Presidential Medal of Freedom "for her exceptional technical leadership, having influenced every major space program from Mercury through the Shuttle program." As Molly Jackson writes, Johnson, now 97 years old, is being recognized for "her groundbreaking contributions to NASA in an era when few African-Americans were working in the field, let alone African-American women." Johnson graduated from high school at 14 and from West Virginia State University (WVSU) at 18 (summa cum laude with degrees in French and mathematics). "At WVSU, Johnson's math talent stood out so much that one mentor, W.W. Schiefflin Claytor--only the third African-American to earn a PhD in math--created a class just for her, studying the analytic geometry of space. But that wasn't enough to open up many post-college opportunities for a young black woman in the 1940s, two decades before President Kennedy's push to integrate NASA, and Johnson wound up as a teacher and homemaker until her NACA [National Advisory Committee for Aeronautics, NASA's predecessor] hire [as a research mathematician]," writes Jackson. At first she was assigned to an all-female unit doing calculations for male engineers, but after the researchers saw her talents they kept her on and gave her more advanced jobs. "Plotting spacecraft trajectories became her specialty." Johnson is quoted as saying she "likes math's constancy, its yes-or-no accuracy."
See "Katherine Johnson was a STEM trendsetter before there was STEM," by Molly Jackson, The Christian Science Monitor, 23 November 2015 and "NASA Langley mathematician named as Presidential Medal of Freedom recipient," by Sarah J. Ketchum, Daily Press, 17 November 2015. See also: NASA's tribute Katherine Johnson: A Lifetime of STEM, and a video interview with Johnson (Maker's Profile).
--- Annette Emerson
On rethinking math education, by Samantha Faria
In this interview, Melanie Matchett Wood, math professor at the University of Wisconsin-Madison, describes how most people truly enjoy math but do not realize it. "They say they don't like math, but they love cooking, or they love quilting, or they love poker. And I say, 'Well, all of those things are math. They're rich in math. You do love math, you're just not identifying all those things as math.'" Matchett Wood's own interest in math was piqued as a middle schooler participating in a Mathcounts competition. "I was asked to solve problems that no one had taught me how to solve," she said. "And that's when I saw how cool math can be--that it wasn't just applying some sequence of steps that the teacher had taught you, but rather figuring out new ways of thinking about things." Today she is a leading researcher in number theory and was recently named as Packard Fellow for Science and Engineering. That does not mean that she is able to uncover answers quickly; she has been working on some problems for years. "You have to try hundreds of things before one of them works…Math is trying of all these new things and maybe one of these will work." When asked about equality in math, Matchett Wood reveals that, "There's still work to be done to keep women in that pipeline into mathematics and, honestly, a lot of related science and technology fields." She believes there must be a shift in cultural norms and a greater effort to rethink how "we understand math."
See "It's Time For Us To Rethink Math, UW Mathematician Says," by Scottie Lee Meyers. Wisconsin Public Radio, 17 November 2015.
--- Samantha Faria
On the graph isomorphism problem, by Allyn Jackson
Is there a polynomial-time algorithm to determine whether two given graphs are the same? This is the graph isomorphism problem, and no one knows how to answer it. It is in the class of problems known as NP: Problems for which a candidate solution can be checked in polynomial time, but for which there is no known polynomial-time algorithm for finding a solution. P is the class of problems for which a polynomial-time algorithm is known. The question of whether P equals NP is perhaps the central question today in theoretical computer science. Most problems known to be in NP have been shown also to be in NP-complete; an NP-complete problem has the property that any NP problem can be reduced to it. The graph isomorphism problem has an unusual status, in that it is one of the few NP problems that researchers suspect is not in NP-complete. In November 2015, Laszlo Babai of the University of Chicago gave a seminar talk in which he described new work on the graph isomorphism problem. This work, the New Scientist article says, shows that "solving graph isomorphism takes slightly longer than polynomial time." A few theoretical computer scientists are quoted in the article saying that, if correct, Babai's result would be an enormous advance. However, Babai himself refused to be interviewed because other researchers have not had a chance yet to check his work. He told New Scientist: "I understand that in the internet age, even a simple seminar announcement can trigger an explosion in the blogosphere, but this is no reason to compromise the process" of peer review.
See "Complex problem made simple sends computer scientists wild," by Jacob Aron. New Scientist, 11 November 2015, "New algorithm cracks graph problem," by Andrew Grant. Science News, 12 December 2015, page 6, and "Landmark Algorithm Breaks 30-Year Impasse," by Erica Klarreich. Quanta, 14 December 2015.
---Allyn Jackson (posted 12/14/15)
Media coverage of the 2016 Breakthrough Prize to Ian Agol, by Annette Emerson
The 2016 Breakthrough Prize in Mathematics was awarded to Ian Agol "for spectacular contributions to low dimensional topology and geometric group theory, including work on the solutions of the tameness, virtual Haken and virtual fibering conjectures." Agol is a professor of mathematics at University of California, Berkeley, currently on sabbatical at the Institute for Advanced Study in Princeton, NJ. (Photo of Ian Agol courtesy UC Berkeley.)
"Agol studies the topology and geometry of three-dimensional spaces, such as our own universe, and has won acclaim for solving five major conjectures by one of the giants in the field, the late William Thurston, a UC Berkeley alum," notes the UC Berkeley news release). "I tend to work very visually, or intuitively, which is something I learned from Thurston," Agol said. "I use my visual cortex to create a shorthand for something that can be very complicated to write down in a precise mathematical way. I try to teach this intuition to my students, too, though eventually you have to convert those visual ideas into equations to show that your intuitions are right."
In Evelyn Lamb's piece, she explains the field of topology: "Agol's field, topology, is the branch of mathematics that pretends all shapes are made of putty or stretchy rubber. It studies those properties that remain the same when the space is squished or stretched, as long as there is no tearing or gluing. You can think of topological properties as the large-scale properties of a space. Geometry, on the other hand, looks at finer properties, those that depend on exactly how the space is put together. Topologists have long had a fairly complete understanding of topology and geometry interact for two-dimensional surfaces, or 2-manifolds. Three-dimensional manifolds are a different story. An appetizing way to understand 2-manifolds and 3-manifolds is to think of a doughnut. The glaze—the two-dimensional donut-shaped surface—is the 2-manifold. The 3-manifold is the whole doughnut, filling and all." She notes that "Agol provided answers to the last of [William] Thurston’s major lingering questions about 3-manifolds.... [His] work gives researchers a way to study these hyperbolic 3-manifolds using surfaces as well.... Specifically, Agol proved the virtual Haken and virtual fibering conjectures." And in response to learning he had won the award Agol said, "Finding out about the prize was never as exciting as the actual moment of thinking I had figured out the virtual Haken question."
The award is US $3 million. As have all five past math laureates, Agol plans to give $100,000 of his prize winnings to support graduate students from developing countries through the Breakout Graduate Fellowships administered by the International Mathematical Union.
The Breakthrough Prize was founded by Mark Zuckerberg and Yuri Milner. "Breakthrough Prize laureates are making fundamental discoveries about the universe, life and the mind," Yuri Milner said. "These fields of investigation are advancing at an exponential pace, yet the biggest questions remain to be answered." This award ceremony was broadcast live on National Geographic Channel November 8, 2015. See video clips, including a video of Agol on "The Beauty of Mathematics."
See media coverage: "By Solving the Mysteries of Shape-Shifting Spaces, Mathematician Wins $3-Million Prize," by Evelyn Lamb, Scientific American, 8 November 2015; "Breakthrough Prizes Give Top Scientists the Rock Star Treatment," The New York Times, 8 November 2015; posts on Reuters and other newswires; coverage in San Jose Mercury News, Science, Forbes, Spiegel Online and news media worldwide; and tweets using hashtag #BreakthroughPrize during and since the televised award ceremony.
--- Annette Emerson (Posted 11/10/15)
On applied mathematics in the Pacific Rim, by Lisa DeKeukelaere
Decades after mathematicians in Europe and North America began working closely with industrial partners, their academic counterparts in Asia are following suit. Mathematicians in New Zealand are working with an agriculture firm to optimize fertilizer use and helping a small start-up fine tune anti-shoplifting software, and an academic study group offers short-term mathematical advice. In Japan, the Institute of Mathematics for Industry had a slow start convincing corporations to accept their interns after starting up in 2011, but using personal connections they eventually placed 50 students who work on projects ranging from financial transactions to computer graphics. Mathematicians in South Korea have been more effective in collaborating with start-ups, and researchers have received grants for projects like modeling how drugs target organs. As more companies come to appreciate the value mathematicians bring, more nations across the Asia Pacific region are joining in.
See "Pacific Rim mathematicians coaxed from their ivory towers," by Dennis Normile. Science, 6 November 2015, page 616.
--- Lisa DeKeukeleare
On math and food, by Claudia Clark
On a recent episode of The Late Show with Stephen Colbert, Colbert did some cooking and calculating with Eugenia Cheng. Cheng is a senior lecturer of pure mathematics at the University of Sheffield and author of the recently published book, How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics. In the segment, when Colbert asks Cheng to explain the title of her book, Cheng responds: "It means taking very simple ingredients like we have here--butter and flour--and doing complicated things with them and making something delicious." To demonstrate an important mathematical concept, Cheng points to a pastry known as a mille-feuille (literally one thousand layers) and asks Colbert, "Could there be one thousand layers here?" She then prepares the puff pastry by wrapping it around 1/2 pound of butter, then alternately rolling it flat and folding it in thirds. The resulting number of layers--which they determine to be the product of 2 and 3^{7}--is indeed larger than 1,000. "Math is not actually just about numbers," Cheng notes. "The principle of this is we used some really tiny numbers--2, 3--and it quickly became a huge number! We made something delicious by the power of exponentials!"
See "Watch Stephen Colbert Try to Wrap His Mind Around the Mathematics of Food," by Daniela Galarza. Eater, 5 November 2015.
--- Claudia Clark
On the latest Ramanujan discovery, by Claudia Clark
In this article, Marianne Freiberger writes about a mathematical surprise discovered recently in the early twentieth-century writings of Srinivasa Ramanujan. This finding, notes Freiberger, "shows that Ramanujan was further ahead of his time than anyone had expected, and provides a beautiful link between several milestones in the history of mathematics." It began with mathematicians Ken Ono and Andrew Granville finding evidence in Ramanujan's manuscript that "Ramanujan had found an infinite family of positive whole number triples x, y, and z" that very nearly satisfy Fermat's famous equation x^{n} + y^{n} = z^{n} for n = 3. These equations are of the form x^{3} + y^{3} = z^{3} ± 1. "Ramanujan had pinned down an infinite family of near-misses of what would be counter-examples to Fermat's last theorem," Freiberger explains. When Ono and his graduate student Sarah Trebat-Leder decided to investigate further, they discovered that Ramanujan had explored the theory of elliptic curves, and had discovered an even more complicated object, which would some 40 years later be named a K3 surface. In fact, writes Freiberger, "his work on the K3 surface he discovered provided Ono and Trebat-Leder with a method to produce, not just one, but infinitely many elliptic curves requiring two or three solutions to generate all other solutions."
See "Ramanujan surprises again," by Marianne Freiberger. Plus Magazine, 3 November 2015.
--- Claudia Clark
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