News from The Division of Physics, Mathematics and Astronomyhttp://pma.divisions.caltech.edu/news/rssen-usTue, 22 Jan 2019 22:31:13 +0000Rana 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. <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="http://www.caltech.edu/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="http://www.caltech.edu/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><span style="font-size: 14px;">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 </span><span style="font-size: 14px;">there is no hidden bias in the factorizations of consecutive numbers into prime numbers.</span></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-82606J. N. Franklin, 1930-2017http://divisions.caltech.edu/sitenewspage-index/j-n-franklin-1930-2017-80482<p>Joel (J. N.) Franklin, who taught mathematics at Caltech for nearly a half century, passed away on November 18 at the age of 87.</p><p>Franklin was born on April 4, 1930, in Chicago to a pair of doctors, J. Nick and Anne Esau. His family moved to Los Angeles when he was 8 years old and, when he turned 16, he changed his last name from Esau to Franklin out of admiration for the intellectual spirit of Benjamin Franklin. That year, he enrolled at Stanford University, where he earned a bachelor's degree in 1950 and a PhD in 1953. His adviser, Dutch mathematician Johannes Gaultherus van der Corput, had been one of the founders and the first director of the Mathematisch Centrum, a mathematical and theoretical computer science research center in Amsterdam. Franklin later did postdoctoral work at the Courant Institute of Mathematical Sciences at NYU.</p><p>After graduate school, Franklin moved to Altadena and, in 1957, began his teaching at Caltech as an associate professor of applied mechanics. He worked closely with Gilbert McCann, professor of applied science, who was one of the early champions of computing at Caltech (and inventor of an analog computer in 1946). Franklin served as a professor of applied science from 1965 to '69, and then as a professor of applied mathematics starting in 1969. He was known for his work on numerical methods, linear and nonlinear computer programming, and problems involving randomness. </p><p>"Joel excelled as a scholar and researcher," says former colleague Dan Meiron, Fletcher Jones Professor of Aeronautics and Applied and Computational Mathematics in the Division of Engineering and Applied Science. "He had a very deep understanding of linear algebra, optimization theory, as well as regularization theory for ill-posed problems. I recall that if any of us in applied math—and the Institute in general—had any questions about matrix theory, linear programming, etc., we could consult with Joel, and he always pointed us to the relevant results often connected to work he had done in the past. He was also a superb teacher. It was routinely the case that we had to find bigger lecture halls to accommodate the large number of students wishing to take his classes."</p><p>Franklin became professor emeritus in 2000. He was the author of a textbook on methods of mathematical economics in 1980, and one on matrix theory in 2000, and was the recipient of Associated Students of the California Institute of Technology (ASCIT) Teaching Awards for the 1977-78 and 1979-80 academic years. In his personal life, Franklin was an accomplished classical pianist. He is survived by his daughter, Holland (Sarah) Franklin, and his grandchildren Benjamin and Kim Seeley.</p>http://divisions.caltech.edu/sitenewspage-index/j-n-franklin-1930-2017-80482Simon Receives Mathematical Physics Prizehttp://divisions.caltech.edu/sitenewspage-index/simon-receives-mathematical-physics-prize-80155<p><a href="http://www.pma.caltech.edu/content/barry-m-simon">Barry M. Simon</a>, the International Business Machines (IBM) Professor of Mathematics and Theoretical Physics, Emeritus, has been awarded the 2018 Dannie Heineman Prize for Mathematical Physics. The prize is administered jointly by the American Physical Society and the American Institute of Physics, and recognizes outstanding publications in the field of mathematical physics.</p><p>Simon was recognized for "his fundamental contributions to the mathematical physics of quantum mechanics, quantum field theory, and statistical mechanics, including spectral theory, phase transitions, and geometric phases, and his many books and monographs that have deeply influenced generations of researchers," according to the award citation.</p><p>"It is a pleasure and honor to get this award, which my advisor—and eight of my co-authors—previously received," Simon says. "As someone who works between mathematics and physics, it is nice to feel validated by the physics community."</p><p>Simon spoke at the International Congress of Mathematics in 1974 and has since given almost every prestigious lecture available in mathematics and physics. He was named a fellow of the American Academy of Arts and Sciences in 2005 and was among the inaugural class of American Mathematical Society fellows in 2012. He has been a fellow of the American Physical Society since 1981. Most recently, Simon received the 2016 Leroy Steele Prize for Lifetime Achievement of the American Mathematical Society. In 2015, Simon was awarded the <a href="http://www.caltech.edu/news/simon-wins-international-mathematics-prize-46655">International János Bolyai Prize of Mathematics</a> by the Hungarian Academy of Sciences, given every five years to honor internationally outstanding works in mathematics, and in 2012, he was given <a href="http://www.caltech.edu/news/caltech-professor-barry-simon-wins-henri-poincare-prize-23607">the Henri Poincaré Prize</a> by the International Association of Mathematical Physics. The prize is awarded every three years in recognition of outstanding contributions in mathematical physics and accomplishments leading to novel developments in the field.</p><p>Simon received his AB from Harvard College in 1966 and his doctorate in physics from Princeton University in 1970. He held a joint appointment in the mathematics and physics departments at Princeton for the next decade. He first arrived at Caltech as a Sherman Fairchild Distinguished Visiting Scholar in 1980 and joined the faculty permanently in 1981. He became the IBM Professor in 1984 and IBM Professor, Emeritus, in 2016.</p>http://divisions.caltech.edu/sitenewspage-index/simon-receives-mathematical-physics-prize-80155Pioneering Physics Show The Mechanical Universe Now on YouTubehttp://divisions.caltech.edu/sitenewspage-index/pioneering-physics-show-mechanical-universe-now-youtube-53331<p>The critically acclaimed television series <em>The Mechanical Universe… And Beyond</em>, created at Caltech and broadcast on PBS from 1985-86, is now available in its entirety on YouTube thanks to the efforts of Caltech's Institute's Information Science and Technology initiative.</p><p>The series was based on the Physics 1a and 1b courses developed by David Goodstein, the Frank J. Gilloon Distinguished Teaching and <span style="font-family: Helvetica; font-size: 13.2px;">Service Professor and Professor of Physics and Applied Physics, Emeritus</span>. It covers topics spanning the scientific revolution begun by Copernicus through quantum theory.</p><p>Each episode opens and closes with Goodstein lecturing to his freshman physics class in 201 E. Bridge, providing philosophical, historical, and often humorous insight into the day's topic. The show also contains hundreds of computer animation segments, created by JPL computer graphics engineer James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.</p><p>Mathieu Desbrun, the John W. and Herberta M. Miles Professor of Computing and Mathematical Sciences, says Caltech was eager to feature the course on its YouTube site because it has been used for decades around the world as a teaching aid, underscoring one of the ways the Institute continues to have an impact disproportionate to its size.</p><p>Although the series was designed as a college-level course, "thousands of high school teachers across the US came to depend on it for instructional and inspirational use," Goodstein says. "The level of instruction in the US was, and remains, abysmally low, and these 52 programs filled a great void."</p><p>The show retains its impact and relevance, partly because "Newton's three laws are still the law of the land," he says—as are other subjects addressed in the series such as relativity, electromagnetic theory, and quantum mechanics.</p><p>Blinn says the series was designed to be rigorous and engaging and used computer animation in a groundbreaking way to visualize mathematical manipulations. Creators of the series referred to the animation as "algebraic ballet," with terms and visual metaphors dancing around the screen to show operations like cancellation and differentiation. "The availability of technology made it so that the developers of the series could see their ideas realized," he says.</p><p>The use of Blinn's computer animations—a rare and expensive technology at the time—made it "legendary," Desbrun says. "<em>The Mechanical Universe</em> is a piece of Caltech history and a source of pride."</p><p>The series can be found online at <a href="http://bit.ly/2gvNAA3">http://bit.ly/2gvNAA3</a>.</p>http://divisions.caltech.edu/sitenewspage-index/pioneering-physics-show-mechanical-universe-now-youtube-53331Practical Mathematics: An Interview with Andrew Stuarthttp://divisions.caltech.edu/sitenewspage-index/practical-mathematics-interview-andrew-stuart-52808<p><em>New Caltech faculty member <a href="http://www.cms.caltech.edu/people/5854/profile">Andrew Stuart</a> is interested in how the current era of data acquisition interacts with centuries of human intellectual development of mathematical models that describe the world around us. As an applied mathematician in the Division of Engineering and Applied Science (EAS), he generates the mathematical and algorithmic frameworks that allow researchers to interface data with mathematical models. His work is informed by—and has applications for—diverse arenas such as weather prediction, carbon sequestration, personalized medicine, and crowd forecasting. Originally from London, Stuart earned his bachelor's degree at Bristol University and then a combined master's/PhD at Oxford University. He worked as a postdoc at MIT in the late '80s, as a lecturer at the University of Bath in England from 1989 to 1992, and then as professor at Stanford University and the University of Warwick in England. He relocated to Southern California this summer. Recently, Stuart answered a few questions about his research and his new life at Caltech.</em></p><h3>What brought you to Caltech?</h3><p>The high quality research in engineering and applied science as well as the high quality of undergraduate and graduate students. I'm excited by the opportunity to develop my mathematical research in new directions, both in terms of applications and in terms of underpinning mathematical methodologies. There's an undeniable beauty to pure mathematics, but what has always driven my interests in mathematics is the potential for diverse applications, and the role of mathematics in unifying these different fields. Caltech provides enormous potential for collaboration in areas of interest to me, in the EAS and Geology and Planetary Sciences divisions for example, and also at JPL.</p><h3>For example?</h3><p>Weather forecasting. Netwon's laws, describing conservation of mass, momentum and energy, in principle have enormous predictive power. But lack of precise knowledge of the initial state of the atmosphere, together with physical effects on scales too small to resolve efficiently on the computer, mean that the "butterfly effect" (in which small changes in complex systems ultimately yield major effects) can lead to poor forecasts. Data provides a potential resolution to this problem, or at least an amelioration of it. Right now we have satellites, aircraft, and weather balloons all collecting vast amounts of data; figuring out how best to use these data can substantially improve the accuracy of our forecasting. A lot of good applied mathematics is about formulating the right problems, as well as finding algorithms for solving them.</p><h3>How did you get into your field?</h3><p>I grew up in an academic household; I saw that it was a challenging, stimulating, and intellectually rewarding career. My dad, who worked at Imperial College in fluid mechanics, loved his job and I was very aware of this. I then developed an excitement for mathematics that grew once I started majoring in the field as an undergraduate student.</p><h3>What are you looking forward to about being in Southern California?</h3><p>The great combination of urban culture and outdoors life. I enjoy cinema, art, reading novels, and hiking. Recently I have been to Kings Canyon and Sequoia, and I have also visited MOCA (Museum of Contemporary Art) Grand Avenue. </p>http://divisions.caltech.edu/sitenewspage-index/practical-mathematics-interview-andrew-stuart-52808Caltech Offers Open Online Course on Quantum Cryptographyhttp://divisions.caltech.edu/sitenewspage-index/caltech-offers-open-online-course-quantum-cryptography-52456<p>This summer, Caltech's <a href="http://eas.caltech.edu/people/5373/profile">Thomas Vidick</a> spent a month delivering a series of lectures about quantum cryptography… to an empty room. On October 9, students around the world will be able to enjoy them.</p><p>Vidick, assistant professor of computing and mathematical sciences in the Division of Engineering and Applied Science, is participating in a massive open online course (MOOC) that will be available, along with two other courses from Caltech, to thousands of students through the <a href="https://www.edx.org/school/caltechx">edX online education platform</a>.</p><p>The class—<a href="https://youtu.be/DERsAtboQ5k">CS/Ph 120, Quantum Cryptography</a>—is cotaught by Vidick and his longtime colleague Stephanie Wehner from QuTech at the Delft University of Technology. Both Vidick and Wehner also will have classroom components to their courses, at their respective institutions.</p><p>Vidick says that he was inspired to teach the course through conversations with his PhD advisor at Berkeley, Umesh Vazirani, who taught a MOOC titled "Quantum Mechanics and Quantum Computation."</p><p>Vidick's course focuses on the ways in which quantum mechanics can be used to create secure lines of communication. Though the concept was first proposed in the 1970s, it has only recently gone mainstream, with the first quantum bank transaction taking place in 2004.</p><p>"It's a hot topic, but there are very few resources for people wanting to go beyond just the basics. Very few schools will even have a quantum cryptography course," Vidick says.</p><p>So far, CS/Ph 120 has 5,500 registered students—small, by the standards of MOOCs, which average 43,000 students, according to a 2014 study by a researcher at the Open University in the United Kingdom. Even so, Vidick expects that just about 200 will stick out the program to the end, given that the average completion rate for MOOCs sits around 6.5 percent.</p><p>For the dozen or so Caltech students and 40 Delft students who will attend in-person, the class will use the "flipped classroom" model, in which the lectures are done online, with time in the classroom spent cementing what the students have learned and diving deeper into the concepts.</p><p>While no prior knowledge of quantum mechanics is necessary, students will need to have a strong grasp of linear algebra, a branch of mathematics central to engineering, in order to follow along, Vidick says. "Making the course accessible does not mean dumbing it down, and the less mathematically inclined might find it challenging," he cautioned in a recent post to his <a href="https://mycqstate.wordpress.com/2016/09/09/coming-to-a-theater-near-you/">personal blog</a>, announcing the course.</p><p>The edX course launches on October 9, although in-class students already have begun meeting, to go over the basics of linear algebra, quantum information, computer science, and cryptography—concepts that will be used throughout.</p><p>Online, students will have access to video lectures, lecture notes, quizzes, and links to additional resources.</p><p>This will be Vidick's first MOOC and his first time teaching quantum cryptography—but he says he is looking forward to the challenge.</p><p>"Every time I finish teaching a class I want to teach it again right away, because it's like <em>'Now</em> I know how to do it,'" Vidick says.</p><p>Students can enroll online at <a href="https://www.edx.org/course/quantum-cryptography-caltechx-delftx-qucryptox">https://www.edx.org/course/quantum-cryptography-caltechx-delftx-qucryptox</a>.</p>http://divisions.caltech.edu/sitenewspage-index/caltech-offers-open-online-course-quantum-cryptography-52456Lifetime of Numbers: Q&A with Barry Simonhttp://divisions.caltech.edu/sitenewspage-index/lifetime-numbers-qa-barry-simon-51679<p><a href="https://www.pma.caltech.edu/content/barry-m-simon">Barry M. Simon</a>, professor of mathematics, emeritus, will be featured on the cover of the<a href="http://www.ams.org/journals/notices/201607/"> <em>Notices of the American Mathematical Society</em></a> on the occasion of <a href="http://www.fields.utoronto.ca/activities/16-17/modern-physics">a special math conference</a> being held this month for his 70th birthday. Simon, who is known as the one of the founding fathers of modern mathematical physics, was recently awarded <a href="/news/simon-receives-lifetime-achievement-award-48745">the 2016 Leroy Steele Prize for Lifetime Achievement of the American Mathematical Society (AMS)</a>.</p><p>Simon has made impactful contributions to the mathematical areas of quantum field theory, statistical mechanics, Schroedinger operators, and the theory of orthogonal polynomials. He has published nearly 400 scientific papers and authored 21 books, including the four-volume textbook series, "Methods of Modern Mathematical Physics," written with Michael Reed in the 1970s. He has also coauthored two popular manuals on how to use Windows computers.</p><p>We recently spoke with Simon about his career, mentoring students, and future goals.</p><h3>You are known as a founding father of mathematical physics. Can you tell us more about the field and how you helped establish it?</h3><p>Modern mathematical physics attempts to establish areas of theoretical physics under the ground rules of rigorous mathematics. There are times that this provides new insights to theoretical physics but, in any event, as my mentor the late Arthur Wightman [Princeton] taught me, intellectual honesty requires the community to understand basic physics at this level of precision.</p><p>I regard the founders of mathematical physics to be a generation before me, notably Wightman, Tosio Kato [UC Berkeley], and David Ruelle [Institut des Hautes Études Scientifiques]. I advanced the field in several ways. My books with Mike Reed served as an introduction to the field and lured a generation of talented people to the area. And I was fortunate enough to be one of the first researchers in a number of areas that allowed me to write seminal papers still widely cited today.</p><h3>When you look back at your decades-spanning career, what do you feel most proud of?</h3><p>The <em>Notices</em> article lists a number of accomplishments and I hesitate to single out only a few, but I'd mention my work on eigenvalue perturbation theory; the work with Francesco Guerra [University of Rome] and Lon Rosen [University of British Columbia] using statistical mechanical methods in Euclidean Quantum Field Theory; my work with Elliott Lieb [Princeton] on Thomas Fermi theory; the work with Jürg Frölich [ETH Zurich], Tom Spencer [Institute for Advanced Study], Freeman Dyson [Institute for Advanced Study], and Lieb on continuous symmetry breaking in statistical mechanics; and my foundational work in ergodic Schoedinger operators and on singular continuous spectrum. Finally, in the past 15 years, I've introduced important new ideas into the spectral theory of orthogonal polynomials.</p><p>I'm also proud of my books and the impact they've had. Perhaps most of all, I am proud of my impact on students, postdocs, and collaborators. There is a special thrill to giving a boost to the careers of young people.</p><h3>The<em> New York Times</em> wrote an article about you winning a math contest when you were 16. Can you tell us about the contest? And did you know then that you wanted to be a lifelong mathematician?</h3><p>This was the exam sponsored by the Mathematical Association of America. Before my year, there were only three perfect scores. I had only one problem wrong but when I was told which one it was, I was dumfounded because I was sure I had it right. The issue was that I interpreted it in a different way from how the exam writers intended it. I appealed and made the case successfully that the wording was ambiguous and thus achieved the second perfect score that year. It was the drama of the appeal that caught the eye of the <em>Times</em>. The actual problem I appealed was part of the article and my brother, Rick, included the text of the article in his contribution to the page of "Barry Stories" put together from my <a href="http://www.math.caltech.edu/SimonFest/stories.html">60th birthday conference</a>.</p><p>At that time, I hardly wanted to be a mathematician. The teacher who had the biggest influence on me in high school was <a href="https://www.youtube.com/watch?v=O_TaUeCLaUU">Sam Marantz</a>, a physics teacher. So while I knew I liked mathematics, I wanted to be a physicist and both my BA, from Harvard, and PhD, from Princeton, are in physics. I knew I wanted to combine my two interests and went to Princeton to study, where I learned that Wightman had done exactly that. For various reasons, all the courses I taught at Caltech were in math, but at Princeton I taught in both departments including the basic undergraduate quantum mechanics. My research in the recent past has focused more on pure math topics so I am perhaps more mathematician than physicist now, but I've always had a joint appointment and been proud of it.</p><h3>Can you help explain to non-mathematicians the importance of math, and specifically of your areas of research?</h3><p>Galileo once said that "the book of nature is written in the language of mathematics." So one importance of mathematics is its central role in all areas of modern science. The remarkable fact is that most mathematicians are not motivated by the applications but by the internal beauty and fascination of the subject. But despite their motivation, what they discover has significant applications in the outside world. For example, something as esoteric as the study of prime numbers is the basis of the encryption you use whenever you connect to your bank's website!</p><h3>In the American Mathematical Society feature story about your career, several colleagues mention how fast you are at writing papers. Can you share your trick?</h3><p>No trick to convey. I've been blessed with a mind that thinks logically and manages to see deep connections so that I am able to write clearly on my first draft. Unlike many scientists I know, I enjoy writing, which makes the process quicker. And I've always worked hard.</p><h3>One of the pictures in the <a href="http://www.ams.org/publications/journals/notices/201607/rnoti-p740.pdf">Notices feature story</a> shows you wearing boxing gloves with Greek letters on them. Is there a story behind this?</h3><p>In about 1995, Caltech decided to redesign its required curriculum. I was the mathematician on the committee chaired by Dave Stevenson, a professor of planetary science. One of things we changed was to decrease the basic calculus classes to one quarter, and I was persuaded to teach the new Math 1a, which I did for six years.</p><p>This course was as much to introduce students to rigorous proof, which many of them hadn't seen in high school, and the centerpiece of that was the use of what are called ε-δ proofs. I was fond of saying "ε and δ are a calculus student's finest weapons." One year, I had an especially lively group of students and, on the last day, when I walked into the auditorium where the class was given there was a pair of boxing gloves on my desk, one with an ε and one with a δ. I've received lots of positive comments from other mathematicians about that picture.</p><p>I have a story about Math 1a that I especially like. I was aware that many students taking that class who taken several years of calculus were offended that we felt they didn't really understand the subject without rigorous proof and they found the course difficult because the approach we thought was essential for them was so foreign to their experience. One day, I was stopped by a student who introduced himself and said: "I'm a senior now and I took Math 1a from you as a freshman. At the time I thought it was the worst course I'd ever taken. I now think it is the best course."</p><h3>How did you get involved in writing Windows manuals?</h3><p>In the mid-1980s as PCs were first coming in, IBM gave a grant to Caltech that let math faculty get IBM XTs. My colleague Rick Wilson, professor of mathematics, emeritus, and I became fascinated with the guts of the machines and developed some expertise in the architecture underlying DOS [disk operating system]. Rick became a first-class assembly-language programmer. This was before there was an active Internet but there was a bulletin board called CompuServe where I met lots of nerds.</p><p>We suddenly found ourselves as shareware authors. We got involved in a CompuServe group trying to set up an API [application programming interface] for resident programs to avoid getting in each other's way. One of the other people involved had done some writing for <em>PC Magazine</em>, at the time the leading computing magazine, and he suggested that I go see them to talk about this API. I did, and by coincidence, one of the editors I met with asked if I knew any mathematics! He was looking for a reviewer for a new program called Mathcad. Soon after, I was the standard reviewer of mathematical software for <em>PC Magazine</em> and many other programs like Visual BASIC.</p><p>Another person, Woody Leonard, I met through CompuServe who knew of my expertise from this writing suggested we write a book about the soon to be released Windows 95. I wouldn't call it a manual—it wasn't so much a detailed how-to as a book to give the reader background for understanding what they are doing. In an era when <em>DOS for Dummies</em> was a bestseller, one wag dubbed our book "Windows for Dummies Not."</p><p>In reference to the phrase popularized during the first gulf war, "mother of all battles," we called our book, <em>The Mother of All Windows Books</em>. We had a version with one of the first CD ROMs of software sold with the book and dubbed that version CD Mom. The books had a fair amount of corny humor and were a lot of fun to write. In all, we published four books together and I had a solo book on Outlook.</p><p>I was struck by the following: A typical math book sells 500–1500 copies. Reed and I were proud of the fact that by 1995, 20 years after it was published, we'd sold almost 15,000 copies of volume one. Well, <em>The Mother of All Windows Books</em> sold 15,000 copies on its first day (and about 50,000 total).</p><h3>You have mentored more than <a href="http://www.genealogy.ams.org/id.php?id=11905">30 graduate students</a>. What is your favorite part of mentoring?</h3><p>When a student starts working for me, one of the first things I give them is a warm-up problem. I get to share in the excitement they feel the first time they realize they can make their own original contributions. And for those that go on in academia, it is always a joy when they get tenure.</p><h3>What is coming up next for you and your research?</h3><p>I officially retired this past June 30. One nice thing I can do without official responsibilities is spend more time in Israel where the majority of my kids and grandkids live. But I expect to continue working. The English mathematician G. H. Hardy once remarked that "young men should prove theorems, old men should write books." While I violated that by publishing the first volume of Reed-Simon at age 26, it does make sense. My five volume, 3,200-page <em>Comprehensive Course in Analysis</em> was published last December, and I've got three book projects in planning stages.</p><p>I also have various research projects under way. Two that excite me are a joint project with two Israeli colleagues that explores some connections between spectral theory and probability theory, and a joint project with two former postdocs that will follow through on our breakthrough last year settling a 40-year-old conjecture in the theory of Chebyshev polynomials. I've also been polishing my website and started an <a href="http://www.math.caltech.edu/simon/selecta.html">online "Selecta</a>," which will include biographical notes and notes on some sets of my papers.</p>http://divisions.caltech.edu/sitenewspage-index/lifetime-numbers-qa-barry-simon-51679