Kavita Ramanan, Richard Schwartz and Joseph Silverman landed the opportunity, considered a career pinnacle for many, to present at next week’s prestigious ICM conference, an event held once every four years.
Professor of Mathematics Joseph Silverman (left) will present “Arithmetic Dynamics: A Survey” at this year's International Congress of Mathematicians. Photos by Nick Dentamaro / Brown University.
PROVIDENCE, R.I. [Brown University] — Every four years, the International Congress of Mathematicians convenes top scholars from across the globe for a series of presentations on every math specialty under the sun. For many invitees, presenting is a career-defining moment — equivalent to an induction to a hall of fame.
This year, as ICM 2022 gets underway from July 6 to 14, no less than three Brown faculty members will have the honor of presenting their work.
Professor of Applied Mathematics Kavita Ramanan will speak about “Interacting Processes on Sparse Random Graphs;” Professor of Mathematics Richard Schwartz will offer “A Survey Lecture on Billiards;” and Professor of Mathematics Joseph Silverman will present “Arithmetic Dynamics: A Survey.”
In addition to their mathematics and applied mathematics faculty appointments, all three scholars are affiliated with Brown’s Institute for Computational and Experimental Research in Mathematics. One of just six federally funded math institutes in the nation, ICERM convenes researchers from across the world for academic programs and activities that advance mathematical understanding.
With ICM 2022 happening virtually this year — the planned in-person gathering in St. Petersburg, Russia, was nixed after Russian military forces invaded Ukraine — ICERM hosted Ramanan, Schwartz and Silverman for advance versions of their ICM addresses, pre-recording them for presentation during the July conference and giving ICERM guests a sneak peek at their ideas and concepts.
ICERM Director Brendan Hassett, also a professor of mathematics at Brown, said he’s proud to see the contributions of Ramanan, Schwartz and Silverman recognized on such a prominent global stage, especially given their impact in helping to build communities of math scholars in their roles at ICERM.
“Really, there are very few institutions that have multiple ICM speakers,” Hassett said. “It shows the tremendous strength that Brown has across both pure and applied mathematics, including in different subspecialties. I’m quite proud that Brown is home to a large mathematics community that cuts across departments, and that we all work collaboratively to create a positive environment for education, training and encouraging mathematics across the country.”
In advance of their ICM 2022 presentations — each of which will be available publicly via the International Mathematical Union’s YouTube channel after the congress ends on July 14 — Ramanan, Schwartz and Silverman shared brief overviews on their topics, and perspective on what it means to have been invited.
I wish I could go back in time and tell younger versions of myself, who struggled with self-doubt and uncertainty, that I would end up at the ICM twice. I have always been kind of an unusual mathematician. I like to work on individual and not necessarily related problems that attract me because of their beauty, simplicity and mystery. These
ICM invitations make me feel like this approach is a valid way of doing mathematics.
Can you share an overview on the work you’ll speak about and its impact in the field?
Ramanan: Large collections of randomly evolving interacting entities model phenomena in a variety of fields, including magnetization in statistical physics, dynamics of spiking neurons in neuroscience, the spread of diseases in epidemiology, and scheduling in communication networks. Quantities of interest — such as the typical dynamics of a single entity or the dynamics of suitable averages of the system — are hard to analyze (or even numerically simulate) due to the complexity and high-dimensional nature of the dynamics. When interactions between particles are weak, then so-called mean-field approximations that overcome the curse of dimensionality were first mathematically justified over a half-century ago and subsequently spawned a huge body of research on related questions that continues to this day. In contrast, it has been an open problem to develop principled approximations justified by rigorous limit theorems when interactions between particles are strong and mean-field approximations do poorly. In the latter setting, not only was no corresponding analytical characterization known, but there was not even a conjecture of what form it might take.
In my talk, I will first survey classical results in the field and then describe my research, which provides a resolution of this open problem and thereby initiates a new line of inquiry. I hope this will not only spur related fundamental research in the theory of stochastic processes, but also the development of corresponding computational algorithms and the analysis of specific models that shed insight into applications.
Professor of Applied Mathematics Kavita Ramanan (right) on her ICM 2022 invitation: "It is heartening to know that the broader mathematical community finds the new perspectives and methods that my research has developed of sufficient significance and interest to warrant an invited talk at the ICM. It provides further impetus for my research, and I hope it also motivates my Ph.D. students, especially those working on related topics."
Schwartz: I’ll speak about mathematical billiards. In a certain sense, this is just like ordinary billiards, played in a pool hall, except that the game here is completely theoretical. The table might have a different shape, and the ball is just a frictionless point moving around. In ordinary billiards, you could imagine trying to figure out what kinds of trick shots are possible; in mathematical billiards, you want to know similar things. For instance, one problem I worked on is whether for every triangular-shaped table, you can find a way to hit the ball so that it makes a perfect repeating path. I made some progress on it but didn't solve it completely.
Another thing I will speak about is outer billiards. This is a game sort of like billiards, except that a ball slides around the outside of the table somewhat like the Moon orbiting the Earth. I solved the main unsolved problem about this kind of system by constructing a particular table for which the corresponding “billiard path” does not stay in a bounded region of the plane. It keeps making wilder and wilder oscillations. This is probably the best work of my career. Some of my ICM talk is a general survey of developments in mathematical billiards and then some of it focuses on the two problems I personally worked on.
Silverman: I'll be speaking about a relatively new field of mathematics called arithmetic dynamics, which I helped to establish in the 1980s. Arithmetic dynamics is an amalgamation of the ancient field of number theory (the study of whole numbers and their relationships to one another) and the field of discrete dynamical systems (how quantities change when a process is repeatedly applied), which evolved in the 1920s. Since starting work in arithmetic dynamics, I have written dozens of articles and a graduate-level textbook, co-organized many conferences and workshops, and supervised 15 Brown Ph.D. theses in the subject.
What is the significance for you, personally and professionally as a Brown scholar, to be invited to present at ICM?
Ramanan: I feel truly honored. It is heartening to know that the broader mathematical community finds the new perspectives and methods that my research has developed of sufficient significance and interest to warrant an invited talk at the ICM. It provides further impetus for my research, and I hope it also motivates my Ph.D. students, especially those working on related topics. Given that circumstances denied us ICM speakers the exciting opportunity to gather together, interact and give our talks in front of a live audience, I am also very grateful to Brendan Hassett and the team at ICERM for arranging a live event at ICERM. I enjoyed learning more about what my colleagues are working on.
It’s a great honor to be invited to speak at the ICM. And it is particularly nice to see that this new field that I helped to found in the 1980s has become a central area of mathematical research.
Schwartz: This is the second time I've been invited to speak at the ICM. The first time was in 2002 in Beijing — I was a professor at the University of Maryland then. So I suppose that the most significant thing for me is that I got invited twice in my lifetime. I wish I could go back in time and tell younger versions of myself, who struggled with self-doubt and uncertainty, that I would end up at the ICM twice. I have always been kind of an unusual mathematician. I like to work on things from scratch and often feel like an outsider. I don't feel like I belong to any particular field and I don't really have a big overreaching goal. I like to work on individual and not necessarily related problems that attract me because of their beauty, simplicity and mystery. I usually take an experimental approach, where I play around on the computer and try to find insights into the problem. So these ICM invitations make me feel like this approach is a valid way of doing mathematics.
Silverman: It’s a great honor to be invited to speak at the ICM. And it is particularly nice to see that this new field that I helped to found in the 1980s has become a central area of mathematical research.
Original source can be found here.
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