News from The Division of Physics, Mathematics and Astronomyhttps://pma.divisions.caltech.edu/news/rssen-usThu, 06 Aug 2020 10:40:04 +0000Math Professor and Students Take 'Random Walk' Togetherhttps://divisions.caltech.edu/sitenewspage-index/math-professor-and-students-take-random-walk-together<p>Some people like to take random walks through the woods, while others might stroll through their own neighborhood. In the world of math, a random walk is in fact more random than this; it would be the equivalent of flipping a coin to decide which direction you would take with each step.</p><p>Recently, Caltech's <a href="https://pma.caltech.edu/people/omer-tamuz">Omer Tamuz</a>, professor of economics and mathematics, along with two of his graduate students, Joshua Frisch and Pooya Vahidi Ferdowsi, and their colleague Yair Hartman from Ben-Gurion University in Israel, solved a long-standing math problem related to random walks. The solution was published last summer in the journal <i>Annals of Mathematics</i>.</p><p>"I remember talking to the students about a realization we had regarding this problem, and then the next morning I found out they had stayed up late into the night and figured it out," says Tamuz.</p><p>"We were very lucky in that this project actually got us the solution we wanted. That's very rare in a math project," says Frisch. "Something like 90 percent of the projects you work on, you are not going to be able to solve. With about 10 percent, you start making progress and work much harder. And even then, you don't always solve those. Part of being a mathematician is getting used to failure. Sometimes you work on something for months and have to give up and go on to the next project."</p><p>Mathematicians imagine random walks in spaces with different dimensions and geometries. In the new study, the Caltech team imagined random walks on "groups," which are objects that can have very diverse geometries. For some groups, the random walks will eventually, after much time has elapsed, converge to a specific direction. In those cases, the walks are said to be path dependent, which means that something that happened in the beginning affects the outcome. Or, in other words, something that happens early on the walk influences where it winds up. But for other groups, the direction of the walks does not converge, and their history does not affect their future.</p><p>"For a random process, is it true that in the long run, everything washes out and whatever happens will happen regardless of what took place earlier? Or is there a memory of what took place before?" asks Tamuz. "Say you have two societies, and one of them makes some technological advancement while the other suffers a natural disaster. Are these differences going to persist forever, or will they eventually disappear and we'll forget that once there was an advantage? In random walks, it has been long known that there are groups that have these memories while in other groups the memories are erased. But it was not really clear which groups have this property and which don't—that is, what makes a group have memory? This is what we figured out."</p><p>The solution, says Tamuz, had to do with finding a "geometric way of describing an algebraic property of the groups." To understand the gist of this, think of a circle. You can describe the circle geometrically (as the set of all points at a given distance from one point), or you can describe it with an algebraic equation. In the case of the random-walk problem, the mathematicians found a new way of thinking about the connections between the geometric and algebraic properties of the groups they were studying.</p><p>"We were actually shocked by how easy it was to solve the problem once we figured out this connection," says Ferdowsi, who explains that though the solution "just flowed out," the team faced a "considerable" delay while he was in his home country of Iran and unable to obtain a visa to come back to Caltech. "In the end, we were delighted to have solved a longstanding open problem in math."</p><p>Frisch says that the big realization they had for this math problem actually grew from a previous problem that was much harder. "I had been bashing my head for a few months on it and couldn't make any progress," he says, "But then we had this eureka idea that applied not only to what we were working on then but also to this more recent problem. It feels really good when you realize, 'Oh my god, this is actually going to work.'"</p><p>The <i>Annals of Mathematics</i> study, titled, "<a href="https://resolver.caltech.edu/CaltechAUTHORS:20190725-090144871">Choquet-Deny groups and the infinite conjugacy class property</a>," was supported by the National Science Foundation and the Simons Foundation.</p>https://divisions.caltech.edu/sitenewspage-index/math-professor-and-students-take-random-walk-togetherNikolai Makarov Honored with 2020 Schock Prizehttps://divisions.caltech.edu/sitenewspage-index/nikolai-makarov-honored-2020-schock-prize<p>Nikolai G. Makarov, the Richard Merkin Distinguished Professor of Mathematics at Caltech, has been awarded the 2020 Rolf Schock Prize in mathematics. The Schock Prizes, which also include awards for logic and philosophy, art, and music, were established and endowed by Rolf Schock, a philosopher and artist who passed away in 1986. They are awarded every three years and decided upon by three committees of the Swedish Royal Academies.</p><p>Makarov is being honored for his "significant contributions to complex analysis and its applications to mathematical physics," according to the award citation. A professor at Caltech since 1991, Makarov, who was born in Russia, received his undergraduate degree from Leningrad University in 1982 and his doctorate from Steklov Mathematical Institute in Leningrad in 1986.</p><p>He has worked in the areas of complex analysis, which investigates functions of complex variables. This field is vital to many branches of mathematics and has numerous applications in the natural sciences and engineering.</p><p>His most famous results concern harmonic measure in two dimensions, stating that the hitting probability distribution on the boundary for Brownian motion in two-dimensional simply connected domains (domains without holes) is one-dimensional. Brownian motion is the random movement of small particles floating in a fluid or gas, which was studied by Albert Einstein in the early 20th century.</p><p>Makarov has also made important contributions in the field of Coulomb gas and growth phenomena in a two-dimensional space. In recent years, he has also produced innovative results in conformal field theory in quantum mechanics, particularly its relationship to complex analysis and probability theory.</p><p>This year's prize ceremony is scheduled to take place on October 19, 2020, at the Royal Academy of Fine Arts in Stockholm.</p><p>Read more about 2020 Schock Prizes at <a href="https://www.kva.se/en/pressrum/pressmeddelanden/schockprisen-belonar-skapandet-av-teorier-konst-och-musik">https://www.kva.se/en/pressrum/pressmeddelanden/schockprisen-belonar-skapandet-av-teorier-konst-och-musik</a>.</p><p></p><p></p>https://divisions.caltech.edu/sitenewspage-index/nikolai-makarov-honored-2020-schock-prizeThe Evolution of Shapehttps://divisions.caltech.edu/sitenewspage-index/evolution-shape<p>New Professor of Mathematics Lu Wang says she first became interested in math in elementary school. Her sixth grade teacher, she says, was passionate about math and taught her that math could be a creative endeavor.</p><p>Today, Wang is just as enamored with math as her teacher. Her speciality is an area known as geometric flow, which involves analyzing the evolution of shapes. The field has known applications in computer graphics and image processing, but Wang says her drive is to "pursue the beauty."</p><p>Wang grew up in Beijing, China. She earned her bachelor's degree in mathematics from Peking University in 2006 and her PhD in mathematics from MIT in 2011. She became a J. J. Sylvester Assistant Professor at Johns Hopkins University in 2011, a Chapman postdoctoral fellow at Imperial College London in 2014, and a tenure-track assistant professor at University of Wisconsin-Madison in 2015. She joined Caltech's faculty in the fall of 2019.</p><h4>What does the field of geometric flow entail?</h4><p>Geometric flow is the study of the shapes of objects and how they can evolve or change. This evolution process is described by partial differential equations. Imagine the shape of a dumbbell with two lobes connected by a long neck. This shape can evolve in ways such that the neck gets smaller and then pinches down to a point, leaving two spheres. The spheres can also shrink down to points. Those points are basically called singularities, and that's when the topology of the shape has changed.</p><p>The topology of an object can be understood by thinking of a donut and a coffee cup. They both have one hole, so they have the same topology. But if one of these objects were pinched to a point, its topology would have changed. I'm interested in understanding how this topological change happens.</p><h4>Are there practical applications for this kind of work?</h4><p>Yes, people use the equations I study for computer graphics and image processing, for example to essentially sharpen the image. Personally, I am motivated by the beauty of math. I think of math as art in some senses. I don't think about how useful my research results are, but focus on the pursuit of beauty and coming up with simple and clean solutions to problems.</p><h4>What are some of the reasons you chose to come to Caltech for math?</h4><p>In the math department here, we have a lot of chances to teach and to interact with young people. Not many departments have these same teaching opportunities. Sometimes, I'll run out of ideas for a problem and get stuck, but I have found that working with young people helps freshen me up and makes me think about things differently. Teaching helps me become unstuck.</p><h4>Can you tell us more about your math teachers growing up?</h4><p>When I was in elementary school, I was very interested in Chinese literature, but then I was lucky to have a very passionate sixth grade math teacher who introduced me to math. My teachers at Peking University were also very good—they have one of the top math programs in the country. My sixth grade math teacher was a woman, while most of my teachers after that were males. One thing I like to do is to inspire more women to go into math. I have participated in various related activities—for example, I co-organized the Women and Non-binary People in Mathematics at Wisconsin (WIMAW) lectures, and I taught in the Institute for Advanced Study Women and Mathematics Program.</p><h4>What does a typical day look like for you?</h4><p>I usually start with checking newly posted articles on the website <a href="http://www.arxiv.org/">www.arxiv.org</a>. I like to listen to classical music, like Chopin, while I am drawing on paper and doing computations. I am constantly communicating ideas with collaborators over email or Skype, or meeting with my students to discuss research projects. Sometimes, when I feel stuck or tired, I like to take a walk around campus or go to the gym.</p>https://divisions.caltech.edu/sitenewspage-index/evolution-shapeDonald S. Cohen (1934–2020)https://divisions.caltech.edu/sitenewspage-index/donald-s-cohen-19342020<p>Donald S. Cohen, Charles Lee Powell Professor of Applied Mathematics, Emeritus, passed away on January 9. He was 85 years old.</p><p>Cohen was one of the first faculty members recruited for Caltech's newly formed applied mathematics program in 1965, starting as an assistant professor of mathematics. He was named associate professor of applied mathematics in 1967 and earned tenure in 1971. Colleagues and former students remember Cohen's outgoing personality, quick wit, and engaging lectures.</p><p>"Don was an inspiring teacher and mentor to several generations of students and colleagues. His energetic presentation of the material he taught, together with his humor and infectious enthusiasm, made his classes memorable," says Guruswami Ravichandran, chair of the Division of Engineering and Applied Science.</p><p>Hans Hornung, C. L. "Kelly" Johnson Professor of Aeronautics, Emeritus, moved into the office next to Cohen upon his arrival at Caltech in 1987, and remembers his colleague and friend as a brilliant teacher who was always ready with a quip or friendly banter. "He was always very funny, with a quick and witty sense of humor," Hornung recalls.</p><p>Cohen's research covered a variety of topics, including early work in the theory of reaction-diffusion equations (which are used to model systems with multiple interacting components, such as chemical reactions). His later research on nonlinear differential equations, pattern formation, stability, and bifurcations had a significant impact on mathematical biology and chemical engineering.</p><p>He formulated models that led to the development of quantitative studies of population dynamics in biological systems, using techniques that he was able to adapt to analyzing the stability of chemical reaction systems. Later, he focused on mathematical models in the materials science of polymer films, studying how the properties of polymers could be tailored to control time release of pharmaceuticals.</p><p>"Don was a truly broad thinker and used his outstanding mathematical modeling skills to provide deep insights into phenomena of interest in materials problems ranging from physical to biological sciences," Ravichandran says.</p><p>Cohen was born in Providence, Rhode Island, in 1934. He earned a bachelor's degree in physics from Brown University in 1956 and a master's degree in mathematics from Cornell University in 1959 for his studies on probability and statistics. He earned a doctorate in applied mathematics from the Courant Institute at New York University in 1962.</p><p>At Caltech, he was a popular teacher who received awards for undergraduate teaching excellence in 1979, 1987, and 1998; in 2000, he was awarded the Richard P. Feynman Prize for Excellence in Teaching. "He had a special ability to make the analysis of even complicated problems seem easy. His playful style in solving problems always entertained, engaged, and challenged students," wrote Thomas Hou, Charles Lee Powell Professor of Applied and Computational Mathematics, of Cohen in <a href="http://eas.caltech.edu/engenious/five/emeritus">an article marking his retirement in 2003</a>.</p><p>"Having Don Cohen as a PhD advisor was a wonderful experience, " says William Kath (PhD '81), now a professor at Northwestern University. Kath recalls the way Cohen was a seemingly bottomless font of humorous stories that would somehow always manage to convey an important life lesson to his students. While advising his own graduate students, Kath often passes these same lessons on to the next generation.</p><p>At Caltech, Cohen was responsible for the creation of Introductory Methods of Applied Mathematics, an in-depth applied mathematics course designed to give undergraduates necessary tools to solve problems in their field, typically those that involve differential equations and special functions. "Don started and built the course from scratch. Ever since, it has been the heart of applied mathematics education at Caltech," Hornung says.</p><p>"It rapidly became one of the largest classes at Caltech, de facto core, even when it was not required by the student's major," says Dave Stevenson, Marvin L. Goldberger Professor of Planetary Science, who has taught the course several times. "Students noted later in their careers that it was a very valuable experience. It could only work at Caltech because of the special nature of our students, and it did not have an exactly comparable counterpart at other institutions."</p><p>Former student Donald Schwendeman (PhD '86), now a professor and department head at the Rensselaer Polytechnic Institute in New York state, credits a meeting with Cohen with helping him decide to attend Caltech as a graduate student: "Cohen's outgoing personality made a big impression on me, but the main point that came through from our first meeting, as I recall, was that Caltech was a place that looked after its grad students." Schwendeman later served as a teaching assistant for Cohen's introductory applied mathematics class.</p><p>"He was a great, no-nonsense teacher," recalls fellow former TA Gregor Kovacic (PhD '90), also now a professor at Rensselaer. "He once told us green TAs how he taught the proof of [a mathematical formula known as] Cauchy's formula: He drew on the blackboard two circles and a pair of close connecting lines between them, with arrows in the opposite directions, and said 'this is the proof.' We were stunned, but yes, it is," Kovacic says.</p><p>Cohen served as the executive officer of applied mathematics from 1988 to 1993 and chair of the Division of Engineering and Applied Science in 1990. He also served as chair of the faculty from 1983 to 1985, and from 1986 to 1987 he chaired the faculty advisory committee of the Caltech Board of Trustees to select Caltech's fifth president. In 1998, he was named Charles Lee Powell Professor of Applied Mathematics. He retired in 2003.</p><p>Beyond Caltech, Cohen served in numerous national organizations, including the Society for Industrial and Applied Mathematics, the American Mathematical Society, and the American Association for the Advancement of Science. From 1993 to 1995, he served as the director of the Center for Nonlinear Studies at Los Alamos National Laboratory.</p>https://divisions.caltech.edu/sitenewspage-index/donald-s-cohen-19342020Caltech Faculty Honored with Breakthrough and New Horizons Prizeshttps://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>https://divisions.caltech.edu/sitenewspage-index/caltech-faculty-honored-breakthrough-and-new-horizons-prizesPhilip Isett Wins Clay Awardhttps://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>https://divisions.caltech.edu/sitenewspage-index/philip-isett-wins-clay-awardSeven from Caltech Elected to National Academy of Scienceshttps://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>https://divisions.caltech.edu/sitenewspage-index/seven-caltech-elected-national-academy-sciencesThe Mathematics of Flowhttps://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>https://divisions.caltech.edu/sitenewspage-index/mathematics-flowSeeking Order in Chaoshttps://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>https://divisions.caltech.edu/sitenewspage-index/seeking-order-chaosCaltech Mathematics Professor Wins 2019 Sloan Fellowshiphttps://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>https://divisions.caltech.edu/sitenewspage-index/caltech-mathematics-professor-wins-2019-sloan-fellowship-85369