News from The Division of Physics, Mathematics and Astronomyhttp://pma.divisions.caltech.edu/news/rssen-usSun, 20 Oct 2019 08:19:31 +0000Caltech Faculty Honored with Breakthrough and New Horizons Prizeshttp://divisions.caltech.edu/sitenewspage-index/caltech-faculty-honored-breakthrough-and-new-horizons-prizes<p>Caltech's Katherine L. (Katie) Bouman has been named a recipient of the 2020 Breakthrough Prize for Fundamental Physics as part of the Event Horizon Telescope (EHT) team that generated the first-ever image of a black hole, while Xie Chen and Xinwen Zhu have each received 2020 New Horizons prizes from the same foundation for their work in physics and mathematics, respectively.</p><p>The Breakthrough Prize, now in its eighth year, is considered the world's most generous science prize. Each Breakthrough Prize is $3 million and the 347 authors of the six EHT papers will divide the award.</p><p>"I was stunned and absolutely thrilled to hear the news," says <a href="http://eas.caltech.edu/people/klbouman">Bouman</a>, assistant professor of computing and mathematical sciences and Rosenberg Scholar in Caltech's Division of Engineering and Applied Science. "I'm so lucky to work with an amazingly talented group of individuals that continues to push the boundaries of science every day. It is such a privilege and an honor to share this award with each one of them."</p><p>Their arresting image of the black hole at the center of the galaxy Messier 87, or M87, captured headlines in April. Given the black hole's distance and the wavelength of light needed to create the image, <a href="/about/news/how-take-picture-black-hole">the EHT team had to build a virtual telescope</a> the size of the earth using radio telescopes around the globe that were synchronized through a network of atomic clocks.</p><p>They then employed multiple classes of imaging algorithms to translate the data they gathered into an image showing the black hole silhouetted against hot gas swirling around it. The EHT's award citation notes that their image "matched expectations from Einstein's theory of gravity: a bright ring marking the point where light orbits the black hole, surrounding a dark region where light cannot escape the black hole's gravitational pull."</p><p>A graduate of the University of Michigan, Ann Arbor, and MIT, Bouman <a href="/about/news/seeing-farther-and-deeper-interview-katie-bouman">joined Caltech's faculty in June</a>, following a postdoctoral fellowship at the Harvard-Smithsonian Center for Astrophysics. "This first black hole image is really just the beginning," she says. "Now that we have access to a laboratory of extreme gravity, we are already thinking of all the ways that we can improve our instrument and algorithms to learn even more. Hopefully soon, we will not just be able to show the world a static black hole image, but a dynamic black hole video of gas spiraling towards an event horizon."</p><p>Caltech's Xie Chen and Xinwen Zhu will receive 2020 New Horizons prizes, which honor promising junior researchers with $100,000 awards for early-career achievements in physics and mathematics.</p><p><a href="http://pma.divisions.caltech.edu/people/xie-chen">Chen</a>, associate professor of theoretical physics at Caltech, is being honored for her "incisive contributions to the understanding of topological states of matter and the relationships between them." She specializes in both the fields of condensed matter physics and quantum information, with a focus on many-body quantum mechanical systems with unconventional emergent phenomena.Her work has potential applications in quantum computing and other related technologies. Chen received her bachelor's degree from Tsinghua University in 2006 and her PhD from MIT in 2012.</p><p>Chen's award is shared with former Caltech postdoctoral researcher Lukasz Fidkowski, now at the University of Washington, along with two others.</p><p><a href="http://pma.divisions.caltech.edu/people/xinwen-zhu">Zhu</a>, a professor of mathematics at Caltech, is being honored for his "work in arithmetic algebraic geometry including applications to the theory of Shimura varieties and the Riemann-Hilbert problem for p-adic varieties." He focuses on the Langlands program—an attempt to unify separate disciplines of math—with applications to both number theory and quantum physics. Zhu found new bridges between the geometric and arithmetic aspects of the Langlands programs and solved outstanding problems in both sides. He received his bachelor's degree from Peking University in 2004 and his PhD from UC Berkeley in 2009.</p><p>Former Caltech postdoctoral researcher Samaya Nissanke, now at the University of Amsterdam, is also a recipient of a 2020 New Horizons Physics prize, along with two others, for "the development of novel techniques to extract fundamental physics from astronomical data."</p><p><a href="https://breakthroughprize.org/">The Breakthrough Prize</a> was founded by Sergey Brin of Google, and Anne Wojcicki of 23andMe; Jack Ma of Alibaba, and Cathy Zhang; Yuri Milner, a venture capitalist and physicist, and Julia Milner; and Mark Zuckerberg of Facebook, and Priscilla Chan. The award will be presented at the eighth annual Breakthrough Prize gala awards ceremony on Sunday, November 3, at NASA Ames Research Center in Mountain View, California, and broadcast live on National Geographic.</p><p>Previous Caltech winners of the Breakthrough Prize include <a href="https://magazine.caltech.edu/esblog/glitz-qubits-main">Alexei Kitaev</a>, the Ronald and Maxine Linde Professor of Theoretical Physics and Mathematics, and <a href="/about/news/john-h-schwarz-wins-fundamental-physics-prize-41536">John H. Schwarz</a>, the Harold Brown Professor of Theoretical Physics, Emeritus, who won the Fundamental Physics prize in 2012 and 2014 respectively. Alexander Varshavsky, the Howard and Gwen Laurie Smits Professor of Cell Biology, <a href="/about/news/caltech-cell-biologist-wins-3-million-breakthrough-prize-life-sciences-41525">received the Breakthrough Prize in Life Sciences</a> in 2014. In 2016, a special Breakthrough Prize in Fundamental Physics was announced <a href="/about/news/ligo-team-awarded-special-breakthrough-prize-fundamental-physics-50657">to honor the LIGO team</a>.</p><p>Previous Caltech winners of the New Horizons Prize include Rana Adhikari, professor of physics, and Maksym Radziwill, professor of mathematics at Caltech, <a href="/about/news/rana-adhikari-and-maksym-radziwill-honored-2019-new-horizons-prizes-84118">who both won in 2019</a>.</p>http://divisions.caltech.edu/sitenewspage-index/caltech-faculty-honored-breakthrough-and-new-horizons-prizesPhilip Isett Wins Clay Awardhttp://divisions.caltech.edu/sitenewspage-index/philip-isett-wins-clay-award<p>Philip Isett, assistant professor of mathematics at Caltech, has been awarded a 2019 <a href="https://www.claymath.org/research">Clay Research Award</a>, presented by the Clay Mathematics Institute for "outstanding achievements of the world's most gifted mathematicians." Isett is receiving the award, together with Tristan Buckmaster of Princeton University and Vlad Vicol of New York University, for the "profound contributions that each of them has made to the analysis of partial differential equations, particularly the Navier-Stokes and Euler equations," according to the award citation.</p><p>The Navier-Stokes equations, proposed in 1822 by Claude-Louis Navier and George Gabriel Stokes, are used to describe fluid dynamics. They are very useful for solving practical problems such as those related to the weather, or the airflow around automobiles or the wings of planes. The Euler equations, named after Leonhard Euler (pronounced "Oiler"), an 18th-century Swiss scientist, are a special case of Navier-Stokes where there is zero internal friction, or viscosity, and are especially interesting for studying turbulence. In 2016, Isett solved a problem related to the Euler equations known as Onsager's conjecture, named after its proposer Lars Onsager, who won the Nobel Prize in Chemistry in 1968.</p><p>"Onsager's conjecture is a problem about the way energy is dissipated in turbulent fluid flow, which is described theoretically by a mechanism called an 'energy cascade,'" says Isett. "Having a confirmation of Onsager's conjecture means roughly that the idea of energy dissipation due to energy cascades is logically consistent with other predictions in turbulence theory about how velocity fluctuates within a turbulent fluid flow."</p><p>Isett received bachelor's degrees in math and economics, with a minor in physics, from the University of Maryland, College Park, in 2008. He earned his PhD in mathematics from Princeton University in 2013. After working at the Massachusetts Institute of Technology as a C.L.E. Moore Instructor and a National Science Foundation postdoctoral scholar, Isett became an assistant professor at the University of Texas at Austin in 2016. He joined Caltech in 2018, and recently <a href="/about/news/caltech-mathematics-professor-wins-2019-sloan-fellowship-85369">won a Sloan Research Fellowship</a>.</p>http://divisions.caltech.edu/sitenewspage-index/philip-isett-wins-clay-awardSeven from Caltech Elected to National Academy of Scienceshttp://divisions.caltech.edu/sitenewspage-index/seven-caltech-elected-national-academy-sciences<p>Two Caltech professors and five Caltech alumni have been elected to the prestigious National Academy of Sciences (NAS). The announcement was made Tuesday, April 30.</p><p><a href="https://www.bbe.caltech.edu/people/dianne-k-newman">Dianne Newman</a> is the Gordon M. Binder/Amgen Professor of Biology and Geobiology and the Allen V. C. Davis and Lenabelle Davis Leadership Chair of Caltech's Center for Environmental Microbial Interactions. Her research focuses on how microorganisms generate energy under conditions where oxygen is scarce—from sediments to soils to chronic infections. She uses geochemical tools to facilitate environmentally informed mechanistic studies of diverse bacteria. Newman is also Caltech's executive officer for molecular biology. In <a href="/about/news/newman-and-orphan-named-macarthur-fellows-52352">2016</a>, she was named a MacArthur Fellow.</p><p><a href="https://www.pma.caltech.edu/people/barry-m-simon">Barry Simon</a> is the International Business Machines Professor of Mathematics and Theoretical Physics, Emeritus. Known as one of the founding fathers of modern mathematical physics, Simon has made contributions to the mathematical areas of quantum field theory, statistical mechanics, Schrödinger operators, and the theory of orthogonal polynomials. In <a href="/about/news/simon-receives-mathematical-physics-prize-80155">2017</a>, Simon received the 2018 Dannie Heineman Prize for Mathematical Physics from the American Physical Society and the American Institute of Physics.</p><p>Newman and Simon join <a href="/about/legacy/historic-awards-honors/national-academy-sciences-members">71</a> current Caltech faculty and one trustee as members of the NAS. Included among this year's new members are five alumni: Christopher Bretherton (BS '80), Edward Callaway (PhD '88), Mark Reid (PhD '76), Bernard Schutz Jr. (PhD '72), and Sue VandeWoude (BS '82).</p><p>The National Academy of Sciences is a private nonprofit organization of scientists and engineers dedicated to the furtherance of science and its use for the general welfare. It was established in 1863 by a congressional act of incorporation signed by Abraham Lincoln that calls on the academy to act as an official adviser to the federal government, upon request, in any matter of science or technology.</p>http://divisions.caltech.edu/sitenewspage-index/seven-caltech-elected-national-academy-sciencesThe Mathematics of Flowhttp://divisions.caltech.edu/sitenewspage-index/mathematics-flow<p>The ways in which water meanders through rivers or makes its way through pipes to your kitchen sink is much more complex than you might think. Mathematicians have been trying to model the flow of water and air for centuries in a field known as fluid dynamics, but according to Philip Isett, a new assistant professor of mathematics at Caltech, the problem is incredibly challenging.</p><p>"Because fluids are ubiquitous in nature, we really have to grapple with understanding them," he says. "Fluids are hard to describe inherently because they exhibit a very chaotic and erratic kind of motion called turbulence."</p><p>Isett received bachelor's degrees in math and economics, with a minor in physics, from the University of Maryland, College Park, in 2008. He earned his PhD in mathematics from Princeton University in 2013. After working at MIT as a C.L.E. Moore Instructor and a National Science Foundation postdoctoral scholar, Isett became an assistant professor at the University of Texas at Austin in 2016. He joined Caltech in 2018, and recently <a href="/about/news/caltech-mathematics-professor-wins-2019-sloan-fellowship-85369">won a Sloan Research Fellowship</a>.</p><p>Isett uses partial differential equations to model fluids; in particular, he studies the Euler equations of fluid dynamics, which date back to their namesake, Leonhard Euler (pronounced "Oiler"), an 18th-century Swiss scientist. Recently, Isett solved a problem related to the Euler equations known as the Onsager's conjecture, named after its proposer Lars Onsager, who won the Nobel Prize in Chemistry in 1968.</p><p>We met with Isett to learn more about fluid dynamics and his love of math.</p><h4>What are partial differential equations?</h4><p>The general field I work in, which is a form of calculus, is called nonlinear partial differential equations. Differential equations are used to measure change. The word "partial" in front of differential equations means that we are calculating more than one variable, such as position and time.</p><p>You can take pretty much any branch of physics and there will be some kind of partial differential equation behind it. In quantum physics, there is the Schrödinger equation; in the general theory of relativity, there are the Einstein equations; and in fluid dynamics, the key equations are the Navier-Stokes and the Euler equations, the latter being what I study.</p><h4>Why are the equations of fluid dynamics important?</h4><p>The Navier-Stokes fluid dynamics equations [proposed in 1822 by Claude-Louis Navier and George Gabriel Stokes] are very useful in a practical sense for solving problems related to all sorts of things like the weather, or the air flow around the wings of planes, where you are predicting what will happen. But in a purely mathematical sense, there are fundamental questions we do not know how to answer about the Navier-Stokes equations. In particular, we do not know if the equations break down and if solutions become so irregular that we cannot use them to predict the future.</p><p>The Euler equations are a special case of Navier-Stokes where there is zero internal friction, or viscosity. They are especially interesting for studying turbulence, because they describe a limiting regime where the internal friction can be ignored. This is a regime where there is a lot of chaotic motion, an example being the turbulence you see in air or water behind a jet or submarine. This turbulence can even happen when you turn on the sink and water comes out very quickly.</p><h4>What are you trying to learn with the Euler equations?</h4><p>We are trying to learn about energy dissipation in these systems. The Euler equations describe a scenario where there is no internal friction, so friction is not what is dissipating the energy but rather something else. Lars Onsager proposed in 1949 that there should exist solutions to the Euler equations that would dissipate kinetic energy without friction and that also would have velocity fluctuations and other properties similar to turbulent flow, thereby linking the concepts of frictionless energy dissipation and turbulence. Building on the work of others and previous work of my own, I was able to prove that Onsager's conjecture is true.</p><h4>What does this mean in a big-picture sense?</h4><p>Solving this problem has theoretical implications because it shows that the idea of energy dissipation independent of internal friction, which is something theorized to occur, is compatible with the predictions about velocity fluctuations and turbulence. This offers some philosophical assurance that the ideas in turbulence theory don't necessarily contradict each other. But also, hopefully the math used to prove these statements is the kind of math that will be truly useful for doing future analyses of the fluid equations.</p><h4>What are you working on now?</h4><p>We are trying to go further than Onsager's conjecture to show that this energy dissipation happens on a local level. There shouldn't be some parts of the fluid where energy is going up and other parts where energy is going down. Energy should be dissipating everywhere. Solving this problem would bring us closer to more realistically describing physical turbulent flow.</p><h4>How did you first get interested in math?</h4><p>I always liked math growing up. I remember when I learned the Pythagorean Theorem. I saw more and more how it was applied to practical problems, for example, to calculate the distance between two points, and I was just so amazed that some person could discover mathematics that could be so useful. I could see that it has a large impact on society.</p><p>When people use the word mathematics, they refer to two different things. On the one hand, there is all the math that people have discovered and know and do, and on the other hand, there is the entire universe of mathematics that is yet to be discovered. I picture the math we know as some kind of surface with lots of winding twists and tangles in it that grow as we learn more, while the math that we don't know can be pictured as a higher-dimensional universe containing that surface. The job of a mathematician is to discover this new math we do not know yet.</p>http://divisions.caltech.edu/sitenewspage-index/mathematics-flowSeeking Order in Chaoshttp://divisions.caltech.edu/sitenewspage-index/seeking-order-chaos<p>Maksym Radziwill, a new professor of mathematics at Caltech, has been fascinated by the question of randomness versus determinism since he was a teenager. "It's phenomenal that you have things which somehow look very chaotic but are in fact predictable," he says, explaining that math can be used to show that seemingly chaotic phenomena have order.</p><p>Radziwill studies number theory, a field that aims to understand properties of integers. He is particularly interested in the interactions of analytic number theory with other fields of mathematics, specifically probability, spectral theory, and harmonic analysis.</p><p>Born in Moscow and raised in Poland, Radziwill earned his bachelor's degree in mathematics from McGill University in 2009 and his PhD from Stanford in 2013. He served as an assistant professor at Rutgers University and at McGill before coming to Caltech in 2018. He recently won the <a href="/news/rana-adhikari-and-maksym-radziwill-honored-2019-new-horizons-prizes-84118">Breakthrough New Horizons in Mathematics Prize</a>.</p><p>We sat down with Radziwill to talk about the meaning of math and how it applies to chaotic systems.</p><h3><b>What made you decide to study number theory?</b></h3><p>What drove me to number theory was its interactions with probability. Let's take the distribution of primes, for example. Individually, it's hard to predict if a number is a prime. For example, if we look at the number 1,027, it's not clear immediately if it's a prime. The same is true for 1,029—we don't know if that's a prime either [both numbers are in fact primes]. But if you look at the global distribution of primes or how often primes come up with subsequent numbers, that's actually predictable. When something is complicated, it often looks random, but that does not mean it is.</p><h3><b>Are you saying that randomness does not exist?</b></h3><p>I am saying that what we perceive as randomness often emerges from having very complicated but deterministic rules. Here's another example: You can generate "random numbers" on a computer, but since computers are deterministic, the numbers simply cannot be random even though they appear to be. It is very hard to tell for an external observer that the numbers come from predictable rules, but they do!</p><h3><b>Can you give an example of how this applies to your research?</b></h3><p>My colleague Kaisa Matomäki from the University of Turku, Finland, and I have done work on the factorization of integers into prime numbers. To factorize means to determine all of the prime numbers that divide a given number. What we have shown, roughly speaking, is that if you pick at random a big interval of numbers, say 10 to the power of 18 numbers, and then compare it to a small interval like 1,000 numbers and do the factorization of each set, then what you'll very likely find is that the way integers factorize in this small interval is similar to the way they factorize in the big interval. There are no hidden biases. So factorization, a purely deterministic process, behaves in a fairly random way.</p><h3><b>Does this finding have any real-world applications?</b></h3><p>This result is reassuring for our cryptographic algorithms, which rely on the assumption that factorization is so complex that it is essentially random. For example, the idea behind the "RSA" public key cryptography protocol is to pick two prime numbers, which are your secret key. When you multiply them, this becomes the public key, which can be shared in the open. Because factorizing this number back into primes is very difficult, one cannot easily recover your private key from the public key. Somebody who has your public key can encrypt a message, but only the owner of the private key can decrypt the message. If there were some hidden biases in the factorization of integers, then that could give a potential attacker a small edge in recovering these two primes that constitute the secret key, thus allowing them to decrypt the message.</p><p></p><p><i>For more information about Radziwill's research, visit his</i> <a href="http://www.its.caltech.edu/~maksym/"><i>website</i></a><i>.</i></p>http://divisions.caltech.edu/sitenewspage-index/seeking-order-chaosCaltech Mathematics Professor Wins 2019 Sloan Fellowshiphttp://divisions.caltech.edu/sitenewspage-index/caltech-mathematics-professor-wins-2019-sloan-fellowship-85369<p>Philip Isett, assistant professor of mathematics at Caltech, has been named a winner of a Sloan Research Fellowship. The fellowships, awarded by the Alfred P. Sloan Foundation, "seek to stimulate fundamental research by early career scientists and scholars of outstanding promise," according to the <a href="https://sloan.org/fellowships/">organization's website</a>. Each year, the Sloan Foundation grants the fellowships to 126 researchers; this year, the awards will come with $70,000 to be spent as the winners see fit.</p><p>Isett works in partial differential equations, focusing on solutions to the incompressible Euler equations of fluid dynamics, which date back to their namesake, Leonhard Euler, an 18th-century Swiss scientist. Fluids are inherently highly complex systems and are hard to precisely analyze with mathematics. Recently, Isett successfully solved a problem related to the Euler equations known as Onsager's conjecture, named after its proposer, Lars Onsager, who won the Nobel Prize in Chemistry in 1968.</p><p>Isett received Bachelor of Science and Bachelor of Arts degrees from the University of Maryland, College Park in 2008 and a PhD from Princeton University in 2013. He then worked at MIT as a C.L.E. Moore Instructor and as a National Science Foundation postdoctoral scholar. He was named assistant professor at the University of Texas at Austin before coming to Caltech in 2018.</p>http://divisions.caltech.edu/sitenewspage-index/caltech-mathematics-professor-wins-2019-sloan-fellowship-85369Rana Adhikari and Maksym Radziwill Honored with 2019 New Horizons Prizeshttp://divisions.caltech.edu/sitenewspage-index/rana-adhikari-and-maksym-radziwill-honored-2019-new-horizons-prizes-84118<p>Two Caltech scientists have been named winners of 2019 New Horizons Breakthrough Prizes.</p><p><a href="http://pma.caltech.edu/people/rana-adhikari">Rana Adhikari</a>, professor of physics, is being honored with the New Horizons in Physics Prize for "research on present and future ground-based detectors of gravitational waves," according to the award citation. He shares the prize with Lisa Barsotti and Matthew Evans (PhD '02) of MIT.</p><p><a href="http://pma.caltech.edu/people/maksym-radziwill">Maksym Radziwill</a>, professor of mathematics at Caltech, is being honored with the New Horizons in Mathematics Prize for "fundamental breakthroughs in the understanding of local correlations of values of multiplicative functions." He shares the prize with Kaisa Matomäki from the University of Turku, Finland.</p><p>The Breakthrough Prizes recognize the world's top scientists in life sciences, fundamental physics, and mathematics. According to <a href="https://breakthroughprize.org/">their website</a>, "the disciplines that ask the biggest questions and find the deepest explanations are the fundamental sciences." In addition to the primary Breakthrough Prizes, worth $3 million, up to three $100,000 New Horizons in Physics Prizes and up to three New Horizons in Mathematics Prizes are given out each year to early-career researchers who have already produced important work in their fields.</p><p>Adhikari is a key member of the Laser Interferometer Gravitational-wave Observatory (LIGO) team, which made the <a href="/news/gravitational-waves-detected-100-years-after-einstein-s-prediction-49777">first-ever direct observation of gravitational waves</a> in 2015. The waves—ripples in space and time—came from a pair of colliding black holes. LIGO has since detected gravitational waves from other cosmic events, including <a href="/news/ligo-and-virgo-make-first-detection-gravitational-waves-produced-colliding-neutron-stars-80082">the collision of two neutron stars</a>. Adhikari's interests lie in fundamental physics, including tests of gravity and quantum mechanics. His group focuses on further improving LIGO's measurements of black holes by exploring the limits of quantum measurements, and using those black holes to measure the shape of the universe.</p><p>Radziwill works in analytic number theory, a field using methods of analysis to understand properties of the integers. He is particularly interested in the interactions of analytic number theory with other fields of mathematics, specifically probability, spectral theory, and harmonic analysis. His recent work with Matomäki further pushes our understanding of the factorization of the integers into prime numbers, by establishing, among other things, that there is no hidden bias in the factorizations of consecutive numbers into prime numbers.</p>http://divisions.caltech.edu/sitenewspage-index/rana-adhikari-and-maksym-radziwill-honored-2019-new-horizons-prizes-84118Caltech Mourns the Passing of Wilhelmus A. J. Luxemburghttp://divisions.caltech.edu/sitenewspage-index/caltech-mourns-passing-wilhelmus-j-luxemburg-83923<p>Wilhelmus Luxemburg, emeritus professor of mathematics at Caltech, passed away on October 2, 2018. He was 89 years old.</p><p>Luxemburg was born on April 11, 1929, in Delft, the Netherlands. In his <a href="http://oralhistories.library.caltech.edu/242/">oral history</a> interview with the Caltech Archives in 2001, he recalled growing up in the Netherlands during World War II, and having to hide out from the Germans in a secret place under the roof of his home, where he and his family would study mathematics and calculus despite living "in fear all day."</p><p><span style="font-size: 14px;">Luxemburg received his BA in 1950 and his MA in 1953, both from the University of Leiden. He earned his PhD from the Delft Institute of Technology in 1955. He joined Caltech as assistant professor of mathematics in 1958, became associate professor in 1960, and professor in 1962, a position he held until becoming emeritus professor in 2000. Luxemburg was also Caltech's executive officer for mathematics between 1970 and 1985.</span></p><p>His main area of study was functional analysis, a branch of mathematics involving infinite-dimensional vector spaces and the connections between them. He also developed methods for applying model theory techniques to conventional mathematics, thereby resolving certain paradoxes of infinitesimal calculus that had been unresolved since the inception of calculus. Luxemburg's most notable work was in the theory of Riesz spaces (partially ordered vector spaces where the order structure is a lattice) and infinitesimals (entities too small to be measured).</p><p>Luxemburg had a love for teaching—he called it "a joy" in his oral history—and working with his math students at Caltech. In his oral history, he said, "Caltech is a unique institution. I don't think there's anything that compares with it in the dedication to the fields that I represented here and what the school is doing. It's not, of course, a huge place like Berkeley and so on, so you know everyone more or less. … And you had all kinds of room to move. … And that is, of course, wonderful—for mathematicians particularly. You have such excellent students, and that makes a big difference."</p><p>Bruce Reznick (BS '73), one of Luxemburg's former students, says, "Professor Luxemburg had an immeasurable mathematical influence on me. I took seven courses from him, each of which I enjoyed immensely." Reznick, who is currently a professor of mathematics at the University of Illinois at Urbana-Champaign, recalls coming back to Caltech to give a lecture 30 years after he graduated and spotting Luxemburg and as well as <a href="http://www.caltech.edu/news/tom-m-apostol-1923-2016-50698">Tom Apostol</a> and <a href="http://www.caltech.edu/news/pioneer-20th-century-mathematics-john-todd-dies-1297">John Todd</a> in the audience. "That was one of my great professional thrills," he says.</p><p>Luxemburg was named a member of the Royal Netherlands Academy of Arts and Sciences in 1974.</p><p><span style="font-size: 14px;">He is survived by a son and daughter and other family members.</span></p>http://divisions.caltech.edu/sitenewspage-index/caltech-mourns-passing-wilhelmus-j-luxemburg-83923Solved! Caltech Researcher Helps Crack Decades-Old Math Problem http://divisions.caltech.edu/sitenewspage-index/solved-caltech-researcher-helps-crack-decades-old-math-problem-83296<p><u><a href="http://iqim.caltech.edu/profile/spiros-michalakis/">Spiros Michalakis</a></u>, manager of outreach and staff researcher at Caltech's Institute for Quantum Information and Matter (<u><a href="http://iqim.caltech.edu/">IQIM</a></u>), and Matthew Hastings, a researcher at Microsoft, have solved one of the world's most challenging open problems in the field of mathematical physics. The problem, related to the "quantum Hall effect," was first proposed in 1999 as one of 13 significant unsolved problems to be included on a list maintained by Michael Aizenman, a professor of physics and mathematics at Princeton University and the former president of the <u><a href="http://www.iamp.org/page.php?page=page_start">International Association of Mathematical Physics</a></u>.</p><p>Like the "millennium" math challenges put forth by the Clay Mathematics Institute in 2000, the idea behind <u><a href="http://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/">these problems</a></u> was to record some of the most perplexing unsolved puzzles in mathematical physics—a field that uses rigorous mathematical reasoning to address physics questions. So far, the problem undertaken by Michalakis is the only one fully solved, while another has been partially solved. Progress made on the partially-solved problem has resulted in two Fields Medals, the highest honor in mathematics.</p><p>"I hope that the solution to this problem will invigorate interest in the field of mathematical physics," says Michalakis. "In mathematical physics, we look for a minimal set of assumptions under which we can show how important phenomena in physics arise. And, as is often the case with proofs of significant problems in math, the solution leads to new ideas and techniques that open the doors to resolving several other important questions."</p><p><strong>Bizarre Electron Behavior</strong></p><p>The original quantum Hall effect was discovered in a groundbreaking experiment by Edwin Hall in 1879 that showed, for the first time, that electric currents in a metal can be deflected in the presence of a magnetic field perpendicular to the surface. Later, in 1980, German experimental physicist Klaus von Klitzing performed Hall's original conductance experiment at a significantly lower temperature and with a stronger magnetic field, only to discover that the electric current was deflected in a quantized fashion. In other words, as the strength of the magnetic field increased, the rise in the electrical conductance of the metal was not gradual or linear, as classical physics predicted, but progressed upward in a step-by-step manner. For this discovery, von Klitzing was awarded the Nobel Prize in Physics in 1985. </p><p>"This is a beautiful problem," says Hastings. "It began with experiments by Hall in the 19th century and by von Klitzing roughly 100 years after Hall. The remarkable thing about the quantum Hall effect is the precise quantization even when there are natural impurities in the material." Hastings says the impurities can affect the path by which current flows through materials. "These impurities are randomly distributed in the material so you might think they would have a random effect on the conductance, but they don't."</p><p>Two years after von Klitzing's discovery, experimentalists Horst Störmer and Daniel Tsui showed something even more baffling: under extreme conditions (even lower temperatures and stronger magnetic fields), the Hall conductance was quantized in fractional multiples of what had been previously observed. It's as if somehow electrons themselves were being split up into smaller particles, each carrying a fraction of the electron's charge. Störmer and Tsui, along with theoretical physicist Robert Laughlin, shared the Nobel Prize in Physics in 1998 for their work on this problem. </p><p>Both the integer and fractional quantum Hall effects indicate that the electrons in these systems are somehow acting together in an unified, global manner, despite their normal tendencies to behave like individual ping pong balls that bounce off each other. Even with all the progress in the field, the question of <em>how </em>the electrons do this lingered.</p><p><strong>A Mathematical Approach</strong></p><p>Michalakis started working on the problem back in 2008 at Los Alamos National Laboratory, where he was a postdoctoral scholar in mathematics. He built his research on pioneering work by Hastings, his adviser at the time, who had developed new mathematical tools for scrutinizing the quantum Hall effect, based on decades of research by others. Michalakis says that reading through all the previous literature proved almost as challenging as solving the problem itself. </p><p>"There was a mountain of research that already existed," he says. "And most of it required advanced knowledge of physics. Coming from a math background, I had to break the problem down into small pieces, each of which I could solve. Basically, I decided to dig under that mountain of knowledge to get to the other side."</p><p>A key to the ultimate solution is topology, which is a way of mathematically describing objects by their shapes. </p><p>"Topology is the study of properties of shapes that don't change when the shape is bent or stretched," says Hastings. "For example, a donut can be stretched into the shape of a coffee cup, but it cannot be turned into a sphere without tearing. Something like this is behind the Hall effect: the conductance isn't changed even though there are impurities in the material."</p><p>The idea that topology was behind the quantum Hall effect was invoked before Michalakis and Hastings became involved, but those researchers had been forced to make one of two assumptions—either that the global view of the mathematical space describing the system was equal to the local view, or that the electrons in the system did not interact with each other. The first mathematical assumption was suspected to be incorrect, while the second physical assumption was not realistic. </p><p>"In a topological state of matter, electrons lose their identity. You get a more spread out, stable, entangled system that acts like a single object," says Michalakis. "Researchers before us realized that this would explain the global properties in the quantum Hall conductance. But they made an assumption that the zoomed-in view was the same as the zoomed-out view."</p><p>Figuring out how to remove both these assumptions is ultimately what stumped the mathematical physics community, spurring them to designate the quantum Hall effect a significant open problem at the turn of the century. </p><p>Michalakis and Hastings succeeded in removing the assumptions by connecting the global picture to the local picture in a novel way. To illustrate their approach, imagine zooming away from Earth. Seeing a sphere without mountains and valleys, you might think you could travel around the planet with no obstacles. But when you come back to Earth, you realize that's not possible—you do have to traverse mountains and valleys. What Michalakis and Hastings' solution does, in a mathematical sense, is to identify an open, flat path that does not encounter any dips or peaks, in essence matching the illusion of what you had perceived globally from above. </p><p>"I used Matt's tools and related ideas from other research to show that such a path always exists and that one could easily find it, if one knew how to look for it," says Michalakis. "The Hall conductance, it turns out, is equal to the number of times that path winds around the topological features of the mathematical shape describing the quantum Hall system. That explains why the Hall conductance is an integer, and why it is so robust against impurities in the physical material. Impurities are like small detours you decide to take from the 'golden' path, as you travel around the world. They won't affect how many times you decide to go around the globe."</p><p><strong>Digesting the Proof</strong></p><p>Michalakis and Hastings's actual proof is of course more complex; the initial proof amounted to 40 pages of mathematical reasoning, but after a painstaking editing process, was whittled down to 30 pages. They submitted their solution in 2009 but it took time for the experts to digest the result, and the proof was not officially published in <em>Communications in Mathematical Physics </em>until 2015.</p><p>Two and a half years after it was published, the community of mathematical physicists officially acknowledged the solution, marking the problem on the <u><a href="http://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/">website list</a></u> as "solved."</p><p>"It took a long time, six years in fact, for the paper to get published, and even longer to be understood and gain the influence and impact that it deserved," said Joseph Avron, professor of physics at Technion-Israel Institute of Technology, writing in the <u><a href="http://phsites.technion.ac.il/avron/wp-content/uploads/sites/3/2018/04/Bulletin-April2018-print.pdf">April 2018 newsletter</a></u> of the International Association of Mathematical Physics.</p><p>Says Michalakis, "The set of assumptions needed to prove the result turned out to be smaller than experts had expected, implying that macroscopic quantum effects, like the quantum Hall effect, should arise in several different settings. This opens new doors and ways of thinking about quantum computing and other quantum sciences."</p><p>The <em>Communications in Mathematical Physics </em>paper describing the solution is titled, <u><a href="http://resolver.caltech.edu/CaltechAUTHORS:20150306-090338269">"Quantization of Hall Conductance for Interacting Electrons on a Torus."</a></u> The research was funded by the National Science Foundation.</p>http://divisions.caltech.edu/sitenewspage-index/solved-caltech-researcher-helps-crack-decades-old-math-problem-83296New Model Shows Where to Improve Power Grids http://divisions.caltech.edu/sitenewspage-index/new-model-shows-where-improve-power-grids-82606<h3><strong>The big question</strong></h3><p>Energy generation from renewable, but fluctuating, resources like solar and wind can add stress to the grid infrastructure, and in particular on specific links within it that can fail and cause brownouts. But how can we know where these vulnerable links are before that happens?</p><h3><strong>The new discovery</strong></h3><p>Researchers from Caltech and Centrum Wiskunde & Informatica national research center in Amsterdam created a mathematical framework that helps to predict where power line failures will occur. In their model, they view a large electricity grid with many renewable inputs as a system of interconnected particles. This allows them to identify which connections between the particles—that is, which power lines in the grid—are most vulnerable to fluctuations in weather patterns, as well as the most likely way that failures will propagate through the network. </p><p>The work—conducted by a team that includes Alessandro Zocca, postdoctoral scholar affiliated with the <a href="http://resnick.caltech.edu/">Resnick Sustainability Institute</a> at Caltech—is described in a paper appearing in Physical Review Letters on June 21. Zocca also works with <a href="http://eas.caltech.edu/people/adamw">Adam Wierman</a>, professor of computing and mathematical sciences in the Division of Engineering and Applied Science, and <a href="http://eas.caltech.edu/people/slow">Steven Low</a>, the Frank J. Gilloon Professor of Computer Science and Electrical Engineering.</p><h3><strong>Why is it important?</strong></h3><p>Renewable energy generation accounts for roughly 15 percent of total energy generation in the United States. Though these sources are cleaner and more sustainable than fossil fuels, they also place a greater strain on the power grid due to inherent, uncontrollable fluctuations in energy generation; solar power can only be generated during the day when it is not cloudy, for example. </p><p>As the U.S. and other nations seek to increase the share of their energy generation that is based on renewable sources, it is important to understand what impact that increase will have on the power grid overall—and how to cost-efficiently compensate for any resulting weaknesses.</p><h3><strong>How it works</strong></h3><p>Particle systems simulate complex structures using a series of interconnected points. Computer animators often use particle systems to simulate explosions, hair, and water. A power grid lends itself to this type of simulation because the grid is actually a network of interconnected points—representing generation stations, transformers, energy consumers, and so on—that are all linked together by power lines.</p><p>Zocca and his colleagues tested their new model using data from the German power grid. The researchers found that they were able to predict which sections of the grid will be likely to fail during periods of high stress—that is, when weather conditions make inputs from renewable sources change unexpectedly. Their model can provide a new framework for detecting vulnerabilities in power grids, allowing those grids to be beefed up or rerouted to better handle stress, preventing power failures as a result.</p><p>The paper is titled <a href="http://resolver.caltech.edu/CaltechAUTHORS:20180621-100145040">"Emergent failures and cascades in power grids: a statistical physics perspective,"</a> and was published by <em>Physical Review Letters</em>. Zocca's coauthors are Tommaso Nesti and Bert Zwart from Centrum Wiskunde & Informatica. Support for this research came from the Netherlands Organisation for Scientific Research and the Resnick Sustainability Institute at Caltech.</p>http://divisions.caltech.edu/sitenewspage-index/new-model-shows-where-improve-power-grids-82606