Mathematics Colloquium
Mathematics Colloquium is held Tuesdays, 4:00pm - 5:00pm in-person in room 310, Linde Hall.
Fall 2025
TUESDAY | Franca Hoffman, CaltechTitle: Covariance-Modulated Optimal Transport and Gradient FlowsAbstract: This talk gives an introduction to the theory of optimal transport and gradient flows in probability space. We present a variant of the dynamical optimal transport problem in which the energy to be minimized is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems up to degrees of freedom given by rotations: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. On the level of the gradient flow, a similar splitting into the evolution of moments and shapes of the distribution can be observed. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of a Gaussian target. |
TUESDAY | Yuri Tschinkel, NYU CourantTitle: Equivariant Birational GeometryAbstract: I will discuss recent results and constructions in higher-dimensional birational geometry over nonclosed fields and in presence of group actions. |
TUESDAY | Amol Aggarwal, Stanford UniversityTitle: Asymptotics for the Toda LatticeAbstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this talk describe several results explaining such asymptotics. The proofs proceed by finding a way to interpret the Toda lattice (under certain random initial data) as a dense collection of solitons and providing a framework to study how these solitons asymptotically evolve in time. In this analysis, Lyapunov exponents (arising from products of random matrices) play an important role. |
TUESDAY | Ruixiang Zhang, University of California - BerkeleyTitle: Where can free waves concentrate?Abstract: Waves are ubiquitous in our daily life. Two best-known linear models are the free wave and free Schrödinger equations, whose simplest forms are very amenable to Fourier analysis. Still, a basic question—how large can a solution be, and where can it be large?—is surprisingly subtle and only partly understood, especially in higher dimensions. Over decades, it transpired that in order to answer this fundamental question, one often needs to understand whether and how much the solution can concentrate on important subsets of ℝn . I will discuss three kinds of such subsets (convex sets, semialgebraic sets and lattices) and their importance based on sample problems. Some of them have nice connections to nearby areas such as number theory, geometry and combinatorics. |
TUESDAY | Sky Cao, MITTitle: Yang-Mills, Probability, and Stochastic PDEAbstract: Originating in physics, Yang-Mills theory has shaped many areas of modern mathematics. In my talk, I will present Yang-Mills theory in the context of probability, highlighting central questions and recent advances. In particular, I will discuss the role of stochastic partial differential equations (SPDEs) in these developments and survey some of the recent progress in this field. |
TUESDAY | Joel Hamkins, University of Notre Dame(DePrima Lecture)Title: What is your Number? Logic Puzzles for MathematiciansAbstract: We are guests at the Midnight Ball, amongst infinitely many friends, and everyone has a natural number written on their forehead. We can see the other numbers, but not our own. The Queen of the Ball is chosen, and she alone is allowed to change her number to whatever she likes. At midnight, we are to shout our best guess for our own number. With advance planning—but no communication after we have our numbers—how well can we do? Find out at the talk, where we shall also explore many other puzzles. We shall meet the blue-eyed islanders and the pirates dividing their treasure. Some solutions rely on the axiom of choice. I shall conclude with the nearly perfect prediction theorem, showing how to predict the current and future values of an unknown (possibly discontinuous) function on the real numbers, with almost-everywhere perfect accuracy, based solely on the history of prior values. |
TUESDAY | Kai Xu, University of California-BerkeleyTitle: TBAAbstract: TBA |