The 2022 Mathematics and Physical Sciences Annual Meeting brought together grantees and scientific partners to exchange ideas through lectures and discussions in a scientifically stimulating environment.
Speakers:
Bhargav Bhatt, IAS and University of Michigan
Greg Bryan, Columbia University
Henri Darmon, McGill University
Aida El-Khadra, University of Illinois
Shafi Goldwasser, Simons Institute for the Theory of Computing
Allan MacDonald, University of Texas at Austin
L. Mahadevan, Harvard University
Emily Rauscher, University of Michigan
Solving polynomial equations modulo a prime number and its powers has a rich history in mathematics going back several centuries. The last decade has witnessed a number of foundational advances in understanding the geometry of these solution sets; these advances have helped resolve longstanding questions in many different areas of mathematics where the p-adic numbers appear, including algebraic geometry, number theory, commutative algebra and homotopy theory. In this talk, Bhatt will give an overview of some of the progress in this area.
After growing up in Bombay, Bhatt moved to the US for his undergraduate education, receiving a BS from Columbia in 2005 and a PhD from Princeton in 2010. Following postdocs at the University of Michigan and the IAS, Bhatt was appointed to the faculty at Michigan in 2014, where he currently serves as the Gehring Professor of Mathematics. Starting in the summer of 2022, he was appointed the Fernholz Joint Professor, jointly between the IAS and Princeton University.
The evolution of our universe is determined by its initial conditions and the physical laws governing its evolution. However, neither of these are open to direct experiment, but must be inferred from observations of distant galaxies. Although enormous progress has been made over the past century, with the discovery of the expansion of the universe and, more recently, the acceleration of that expansion, our understanding of the evolution of the universe and the structure within it remains incomplete. This is set to change in the near future, with many new large telescopes and space missions promising a deluge of data, but current models will only be able to use a small fraction of this information. In this talk, Bryan will provide an overview of a large collaborative effort to bring together a range of experts from cosmology, galaxy formation, machine learning and statistical inference in order to create new tools and techniques that will be able to make full use of all of the observations. In particular, we hope to — conceptually — take the picture of the current universe that is now coming into focus and run it backwards in time to create a map of the initial conditions in the early universe. Bryan will highlight a few areas of recent promise.
Greg Bryan is a professor of astronomy at Columbia University. He received a Ph.D. in astrophysics at the University of Illinois at Urbana-Champaign in 1996 and held positions at Princeton, MIT and Oxford before joining the faculty at Columbia in 2004. He is a recipient of a Princeton Lyman Spitzer Fellowship, a Hubble Fellowship, an NSF CAREER Award and the Leverhulme Trust Prize. His primary research focus involves the use of large-scale computational hydrodynamics and computational models to better understand astrophysical systems in a cosmological framework. He has applied such techniques to study the generation of large-scale structures in the universe, the formation of X-ray clusters, the evolution of galaxies and the birth of the first stars in the universe. He has also carried out numerical simulations used to generate visualizations for the Oscar-nominated IMAX film Cosmic Voyage, as well as planetarium shows at the American Museum of Natural History.
Henri Darmon
McGill University
Complex Multiplication and Kronecker’s Jugendtraum: A Non-Archimedean Perspective View Slides (PDF)
Vigorously developed in the 19th century by mathematicians like Eisenstein and Kronecker, the theory of complex multiplication asserts that the values of modular functions at imaginary quadratic arguments of the Poincaré upper half plane belongs to abelian extensions of the relevant quadratic field and, indeed, generate essentially all such extensions.
Generalizing this statement to other base fields is the 12th in Hilbert’s celebrated list of open problems for the 20th Century and remains poorly understood in the 21st century. This talk will describe a possible non-Archimedean approach to extending the theory of complex multiplication to real quadratic fields, in which modular functions are replaced by objects called “rigid meromorphic cocycles.”
Henri Darmon was born in Paris in 1965. He did his undergraduate studies at McGill and wrote a Ph.D. at Harvard under the supervision of Benedict Gross in 1991. After a postdoctoral stint at Princeton, he joined the faculty at McGill in 1994, where he is currently a James McGill Professor in the Department of Mathematics and Statistics. His research interests revolve around the theory of elliptic curves, modular forms and their associated L-series, with a special emphasis on explicit methods and computational issues.
More than eighty years after the muon, the heavy sibling to the electron, was first identified, it may become a window to discovering new physics — a central goal of the high energy physics community. In April 2021, the Fermilab experiment announced, to worldwide media attention, a new measurement of the muon’s magnetic moment with an exquisite precision of 352 parts per billion, sharpening the longstanding tension between experiment and theory to a tantalizing 4.2 standard deviations. The experimental measurements will continue to improve with the goal of reducing the uncertainties by a factor of four. The theoretical calculations of the muon’s magnetic moment must account for the virtual effects of all particles and forces within the Standard Model, where effects coming from virtual hadrons, governed by the strong interactions, are the by far largest sources of theory uncertainty. El-Khadra will discuss the ongoing interplay between theory and experiment that is essential to unlocking the discovery potential of this effort.
Aida El-Khadra received her Ph.D. in 1989 from the University of California, Los Angeles, after receiving her diploma from the Freie Universität, Berlin, Germany. She held postdoctoral research appointments at Brookhaven National Laboratory, Fermi National Accelerator Laboratory and the Ohio State University before joining the faculty at the University of Illinois in 1995. She is a fellow of the American Association for the Advancement of Science, the American Physical Society, a recipient of a Sloan Research fellowship and a Department of Energy Outstanding Junior Investigator Award. In addition to several research and teaching awards from the University of Illinois, she has also been named a Fermilab Distinguished Scholar.
El-Khadra works on topics in theoretical high energy physics, where she has made significant contributions to the development of lattice quantum chromodynamics. Her focus is on precision calculations that are needed to interpret measurements in high energy experiments. She is a leader of the Fermilab Lattice and MILC collaborations and is chair of the steering committee of the Muon g-2 Theory Initiative.
Shafi Goldwasser
Simons Institute for the Theory of Computing
Simons Institute for the Theory of Computing: Past and Present View Slides (PDF)
Established in 2012, the Simons Institute for the Theory of Computing brings together researchers in theoretical computer science and related fields, with emphasis on the next generation of outstanding young scholars, to explore research problems within the context of research programs dedicated to different topics in the field and adjoining areas. Since opening its doors to visitors in 2013, the Simons Institute has hosted over 2,000 long-term visits in 28 research programs and had 19,000 workshop registrations. Whereas the institute’s early work focused on an atomized model of discrete programs, the institute is increasingly emphasizing overarching themes and is exploring innovative ways to connect interrelated programs. These themes are reflective of developments in the field — including the emergence of machine learning, cryptography and privacy, and quantum computing as central topics. As the institute enters its second decade, and as public understanding and use of algorithms continues to increase, the institute anticipates including programming which will have impact on societal challenges.
Shafi Goldwasser has had tremendous impact on the development of cryptography and complexity theory. Starting with her thesis on “semantic security,” she laid the foundations of the theory of cryptography. She created rigorous definitions and constructions of well-known primitives such as encryption schemes (both public and private key versions) and digital signatures, and of new ones that she introduced, such as zero-knowledge interactive proof systems invented with Micali and Rackoff. Continuing her work on interactive proofs that allow a probabilistic polynomial time algorithm to verify mathematical proofs via interaction with a powerful prover, Shafi and her co-authors extended the notion of interactive proofs to two-prover systems. The original motivation was cryptographic, but they turned out to be of great significance in complexity theory, paving the way to the equivalent formulation of probabilistically checkable proofs (PCP). The expressive power of two-prover systems is huge (non-deterministic exponential time). Furthermore, Shafi and her co-authors showed the connection between a scaled-down variant of these systems and the hardness of approximation results for NP-hard problems, which led to the PCP theorem. On the algorithmic front, a problem of great significance is that of recognizing (and generating) prime numbers. Shafi and Kilian designed efficient probabilistic primality provers, which output short proofs of primality, based on the theory of elliptic curves. Together with Goldreich and Ron, Shafi originated the field of combinatorial property testing, devising a class of sub-linear algorithms to test properties in dense graphs.
Allan MacDonald
University of Texas at Austin
The Magic of Moiré Materials
Two-dimensional crystals that are overlaid with a difference in lattice constant or a relative twist form a moiré pattern. In semiconductors and semimetals, the low-energy electronic properties of these systems are described by Hamiltonians that have the periodicity of the moiré pattern, opening a strategy to make artificial two-dimensional crystals with lattice constants on the ten-nanometer scale. MacDonald refers to these artificial crystals as moiré materials. Because of their large lattice constants, the band filling factors of moiré materials can be tuned over large ranges without introducing chemical dopants simply by using electrical gates. Moiré materials can be used to flexibly simulate the physics of real atomic scale crystals and to create new states of matter that raise interesting mathematical questions.
MacDonald will survey progress that has been made in understanding the low-temperature properties of the first moiré materials — twisted graphene systems in which electron velocities vanish at discrete magic angles and two-dimensional transition-metal dichalcogenide stacks that simulate atomic scale Hubbard model physics — and speculate on future directions.
Allan MacDonald is a theoretical condensed matter theorist whose work focuses on predicting new electronic properties in new materials and on explaining poorly understood observations related to the quantum physics of interacting electrons in materials.
Among other topics, he has made theoretical contributions to theories of the integer and fractional quantum Hall effects, spintronics in metals and semiconductors, topological Bloch bands, correlated electron-hole fluids and exciton and polariton condensation, and two-dimensional materials. In 2010, MacDonald predicted that it would be possible to realize strong correlation physics in graphene bilayers twisted to a magic relative orientation angle, foreshadowing the rise of twistronics. His recent work is focused on anticipating new physics in moiré superlattices and on achieving a full understanding of the magic-angle graphene and transition-metal dichalcogenide systems.
Dr. MacDonald is a member of the American Academy of Arts and Sciences and the U.S. National Academy of Sciences and has been awarded the Herzberg Medal (1987), Buckley Prize (2007), Ernst Mach Honorary Medal (2012) and Wolf Prize (2020).
Understanding the emergence of functional organismal geometry from genetics is one of the great challenges of biology. Using examples from across the tree of life, Mahadevan will discuss the geometry and physics of multicellular morphogenesis. Along the way, he will also discuss aspects of morphometrics — the quantification of biological shape, morphogramming — the control and design of bioinspired shape, and how biological self-organization raises new questions in physics and mathematics.
L. Mahadevan was born in India and, after his undergraduate degree at IIT-Madras, moved to the US for graduate school, eventually getting his Ph.D. at Stanford University. Following visiting appointments at the University of Illinois and the University of Chicago, he started as a faculty member at MIT before moving to Cambridge University as the inaugural Schlumberger Professor of Complex Physical Systems. He has been at Harvard since 2003 as the Lola England de Valpine Professor of Applied Mathematics, Physics, and Organismic and Evolutionary Biology. Since 2017, he has also served as a Faculty Dean of Mather House at Harvard College, living and learning with a community of more than 400 students.
Mahadevan’s research attempts to understand motion and matter at the observable and experiential scale of “middle earth” by integrating experiments, theory and computation. Areas of interest include the patterns of shape and flow of inanimate matter and the dynamics of sentient living matter that can self-organize, perceive and act. Specific questions to which he has contributed answers range from the mechanical basis for biological morphogenesis to the inverse design of origami and kirigami tessellations, and from the probabilistic nature of geometric cognition to the collective behavior of super organisms. Mahadevan is a MacArthur Fellow, a Fellow of the Royal Society and a Simons Investigator.
Emily Rauscher
University of Michigan
The Peril and Promise of Three-Dimensional Exoplanets View Slides (PDF)
Much to the consternation of spherical cows everywhere, planets are three-dimensional objects. This is even true for exoplanets because, even though they are generally unresolvable in the glare from the stars around which they orbit, when we interpret atmospheric measurements of these worlds it may be inappropriate to pretend that all regions of the planet have the same temperature and composition. In fact, we may trick ourselves and retrieve biased values when we use 1D models to interpret atmospheric characterization measurements. This is particularly true for the high signal-to-noise type of exoplanet known as “hot Jupiters” because of the intense stellar heating they receive on their permanent day sides. In this talk, Rauscher will discuss how we can turn this challenge into an opportunity, using 3D models of exoplanet atmospheres to uncover the influence of complex physics in different types of observations and thereby empirically constrain the inherently 3D structure of these planets. Rauscher will review how we measure exoplanet atmospheres and discuss her group’s 3D modeling work, highlighting connections to observations. The necessity of a 3D approach to exoplanet atmospheric characterization is becoming more compelling as we move into the era of the James Webb Space Telescope (JWST) and extremely large telescopes, with their upcoming exquisitely detailed measurements.
Emily Rauscher is a theoretical astrophysicist who studies exoplanets, particularly a type called hot Jupiters. She uses a 3D atmospheric circulation numerical code to model the wind and temperature structures in exoplanet atmospheres, both to understand the exotic physical processes at play and to investigate how these complex properties influence various types of atmospheric measurements. Her group’s 3D model has been used to study physical processes such as radiatively active cloud species that form and dissipate as the simulation runs and the influence of magnetic drag when the winds become weakly thermally ionized. Rauscher has studied how orbital phase curves can constrain the day-night differences around a planet, has pioneered eclipse mapping as a method to resolve 3D maps of exoplanet daysides and is a leader in identifying how 3D atmospheric properties (including winds and rotation) influence high-resolution spectroscopic measurements. She is a member of the science teams for multiple JWST programs and the two high-resolution instruments being built for the European Extremely Large Telescope (METIS and ANDES).
Dr. Rauscher received her B.A. in Physics and Astrophysics from the University of California, Berkeley, and her Ph.D. in Astronomy from Columbia University. She received the NASA Sagan Postdoctoral Fellowship, spending the first two years in the Lunar and Planetary Laboratory at the University of Arizona before moving to the Department of Astrophysical Sciences at Princeton University, where she also held a Lyman P. Spitzer Jr. Postdoctoral Fellowship. She arrived in the Astronomy Department at the University of Michigan as a President’s Postdoctoral Fellow in 2014, began her faculty appointment in 2015 and is currently an Associate Professor. Her awards include being named a Cottrell Scholar, receiving the University of Michigan Class of 1923 Memorial Teaching Award and, most recently, being named a Simons Fellow in Theoretical Physics.
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