Abstract: We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow- and diffusion-based generative models by deriving continuous-time normalizing flows and score-based models through different choices of particle dynamics and cost functions. We study the mathematical structure and properties of each generative model by examining their associated MFG optimality conditions, which consist of coupled forward-backward nonlinear partial differential equations (PDEs). We present this framework as an MFG laboratory, a platform for experimentation, invention, and analysis of generative models. Through this laboratory, we show how MFG structure informs new normalizing flows that robustly learn data distributions supported on low-dimensional manifolds. In particular, we show that Wasserstein proximal regularizations inform the well-posedness and robustness of generative flows for singular measures, enabling stable training with less data and without specialized architectures. We then briefly discuss applications of these principled generative models to scientific computing, including simulation-based inference and operator learning.
Abstract: Suppose we have an orientable Riemannian surface M with genus G and a single boundary component, and we also have a Riemannian disc D such that points on its boundary are closer together than the corresponding points on the boundary of M. How does the area of D compare to the area of M? By extending methods of S. Ivanov, we will prove an inequality relating these two quantities. We will also discuss the relationship between this and Gromov's filling area conjecture.
Abstract: Relative Langlands duality, formulated by BZSV, predicts some duality properties between 'periods' and 'L-functions'. I will briefly introduce the formulation of global numerical problems, and give several examples, and how this work fits into the framework of BZSV.
Abstract: In this talk, we present an overview of some of our recent works on the differentiable programming paradigm for learning, control, and inverse modeling. These include using dynamics-inspired, learning-based algorithms for detailed garment recovery from video and 3D human body reconstruction from single- and multi-view images, to differentiable physics for robotics, quantum computing and VR applications. Our approaches adopt statistical, geometric, and physical priors and a combination of parameter estimation, shape recovery, physics-based simulation, neural network models, and differentiable physics, with applications to virtual try-on and robotics. We conclude by discussing possible future directions and open challenges
Abstract: Serotype replacement - an increase of non‑vaccine serotypes after vaccination - is usually attributed to explicit competition in transmission models. I will show that even without such competition terms, standard assumptions on how many serotypes a host can carry (multi‑colonization) can generate strong, competition‑like effects. Using SIS‑type models with two and three serotypes, I derive explicit coexistence formulas where each serotype’s equilibrium prevalence equals its single‑serotype endemic level minus a term reflecting multi‑colonization constraints. When triple colonization is allowed, serotypes decouple and a monovalent vaccine against one serotype does not affect others. When hosts can carry at most two serotypes, double‑colonized hosts become structurally unavailable to the remaining serotype, creating “implicit competition” and potential replacement.
Abstract: Two 20 minute talks. The lack of creativity seen in ChatGPT is not due to the nature of AIs. Game-playing AIs like AlphaZero have an architecture that promotes creativity.
Abstract: The mapping class groups of locally finite infinite graphs are non-Archimedean topological groups that directly generalize the outer automorphism groups of free groups. I will discuss the sphere complex associated with a graph and outline my own work as well as joint work with Hill-Kopreski-Rechkin-Shaji which together show that the automorphism group of the sphere complex is naturally isomorphic to the mapping class group of the associated graph. With whatever time is remaining, I'll introduce a sort of "metaconjecture" which allows one to translate many results about mapping class groups of surfaces to mapping class groups of graphs, and list some consequences.
Abstract: I will present recent work on the analysis of a class of parameterized linear nonlocal problems. The parameter represents the degree of nonlocality, where both the corresponding function spaces and solutions for given data depend on this value. As the parameter varies, the nature and type of the equations may change, for example, from a sequence of nonlocal equations to a limiting local equation.
Focusing on scalar-valued fractional equations and the strongly coupled systems of nonlocal equations derived from the bond-based model of peridynamics, we prove the variational convergence of the models and establish rigorous criteria for compactness. Furthermore, we analyze the behavior of solutions as a function of the nonlocality parameter. We also develop a numerical approximation framework that provides solutions for each parameter while remaining compatible with the asymptotic continuity property of the problem. Specifically, the framework guarantees unconditional convergence with respect to both modeling and discretization parameters to the solution of the corresponding limiting problems.
Abstract: Fluid dynamics is a very old field which is at the same time classical physics and pure mathematics. Some of the most important problems on the physical side, including the emergence and persistence of turbulence, are intimately connected to mathematical issues of singularity formation and propagation for the Euler and Navier-Stokes equations. On the mathematical side, there have been recent breakthroughs in understanding exactly these issues for both incompressible and compressible fluids. This workshop aims to bring together both senior and junior researchers in this area to explain recent progress and highlight future directions of promise.
Abstract: The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for vector bundles and higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. In this talk, we will study Brill-Noether loci for vector bundles on the projective plane in the case where the number of sections is close to the largest possible number. When the number of sections is very large, Brill-Noether problems are all "trivial"--the Brill-Noether loci are either empty or the entire moduli space. As the number of sections decreases, we find that there is a "first" nontrivial Brill-Noether locus, and we discuss its geometry.
Abstract: I will talk about diffusions with multiple invariant manifolds of varying dimensions and thus with varying degrees of degeneracy. Each of the invariant manifolds may carry an invariant measure. The long-term statistical properties of such a system may be governed by one or more such invariant measures and transitions between them. The resulting nonergodic intermittent averaging cannot be described by the classical ergodic theory. This is joint work with Renaud Raquepas and Lai-Sang Young.
Abstract: A nearly hundred-year old question by Fatou asks for a synthesis of the following two kinds of holomorphic dynamical systems under a common framework of holomorphic correspondences on the Riemann sphere: (a) Kleinian groups acting on the Riemann sphere (b) iteration of complex polynomials on the Riemann sphere. Sullivan's dictionary gave us a way of translating techniques from one of these fields to give results in the other. In a relatively recent development, building on Sullivan's dictionary, a bridge has been built between these two classes in the spirit of Bers' simultaneous uniformization theorem. New holomorphic dynamical systems on the Riemann sphere have thus been discovered that arise as combinations or matings of Kleinian groups and polynomials. In some cases, these single valued matings give rise to multi-valued algebraic correspondences on the Riemann sphere, partially fulfilling Fatou's dream. A particular consequence of these constructions is an analog of the compactness theorem for Bers slices of punctured sphere groups. In 1982, Thurston posed a number of questions that guided the development of the theory of Kleinian groups for the next 3 decades. With the above analog of Bers compactness in place, many of these questions reincarnate themselves in this new context. We will survey some of these developments and questions. This is joint work with Yusheng Luo and Sabyasachi Mukherjee.