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.

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.

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.