Abstract: This workshop gathers interdisciplinary researchers (including mathematicians, epidemic modelers, and behavioral scientists) to explore the interplay between human behavior and disease dynamics and how to translate principles of coupled dynamics for public health benefit.
Abstract:Â In my recent work on a geometric proof of the endoscopic fundamental lemma forspherical Hecke algebras, there are many new features not present in its Lie algebra analogue originally proved by B.C.~Ng\^o.
One of such new features is a combinatorial analogue of the fundamental lemma, called the asymptotic fundamental lemma (AsFL). Not only does it imply the original fundamental lemma when combined with results about multiplicative Hitchin fibrations, it also connects the mysterious restriction functor for Kashiwara crystals with the transfer factor, hinting at its potential role in functoriality in general.
In the first talk of this two-talk series, I will explain what fundamental lemma is and the motivation behind. I will then discuss how to translate it into geometry, and how it leads to the asymptotic fundamental lemma.
Abstract: In this talk, we survey recent continuous-time deep learning approaches, present applications in high-dimensional mean field games and optimal control, and discuss efficient training and quantization techniques. We demonstrate how continuous-time deep learning models can address high-dimensional mean field games and optimal control problems, extending their application to state spaces with dimensions reaching the hundreds. For these applications, we leverage neural ordinary differential equations (neural ODEs) in combination with scalable Lagrangian PDE solvers to mitigate the curse of dimensionality, optimizing a neural network representation of the value function with penalty terms that enforce Hamilton-Jacobi-Bellman (HJB) equations—eliminating the need for pre-computed training data. We conclude with ongoing research on improving deep neural network training efficiency through mixed-precision computation and modified Gauss-Newton algorithms. We reduce model size and computational cost by dynamically adjusting the floating point precision and leveraging advanced automatic differentiation.
Abstract: Given a $3$-dimensional manifold $X$, and a positive real number $k$, we define a $k$-surface in $X$ to be an immersed surface of constant extrinsic curvature equal to $k$ which is complete with respect to the sum of its first and third fundamental forms. The theory of such surfaces was revolutionized by François Labourie in a series of papers from the late 80's to the early 2000's which revealed how Gromov's theory of pseudo-holomorphic curves may be fruitfully applied to their study. In this talk, I will discuss my own recent contributions.
Abstract: In joint work with A. Seppi and J. Toulouse we solve the Plateau problem for complete, maximal spacelike submanifolds of pseudohyperbolic space $\Bbb{H}^{p,q}$, with interesting applications to the study of Anosov representations in $\text{O}(p,q+1)$. In this talk, I will discuss some of the analytic aspects of our construction..
Abstract: In this talk, we discuss some applications of model theory to machine learning. In particular, we explore PAC-learning and differential privacy. It has recently been shown (Alon, Bun, Livni, Malliaris, Moran) that a concept class admits a differentially private PAC-learning algorithm if and only if it has finite Littlestone dimension. That is, if and only if it is a definable family in a model of a stable theory. The best known bounds for the sample complexity of a differentially private PAC-learning algorithm on a concept class with Littlestone dimension d is roughly O(d^6) (Ghazi, Golowich, Kumar, Manurangsi). In our current work, we are looking for an improvement to this bound. Towards this end, we are examining special cases of stable theories, like equational theories (as was recommended to us by Artem Chernikov).
This work is joint with Alexei Kolesnikov, Miriam Parnes, and Natalie Piltoyan.
Abstract: By a classical result of Diedrich and Fornaess for a bounded pseudoconvex domain in C^n with smooth boundary one can find a smooth defining function \rho and b>0 such that -(-\rho)^b is plurisubharmonic. Such a b is called a Diederich-Fornaess exponent. We will give a quantitative version of this result and also present the situation for the worm domains, generalizing a result of B. Liu.
Abstract:Â This is the second talk in a two-talk series. Continuing from theprevious one, I will present some examples of asymptotic fundamental lemma and explain how Kashiwara crystals and Langlands--Shelstad transfer factor fit in the picture.
Abstract:Â It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\mathbb{Z}^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $d0$ must be false if $\chi=0$. Â A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false. Â These consequences support the idea that when $\chi=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.
Abstract: The problem of classifying smooth, closed, highly-connected n-manifolds (for n \geq 5) up to diffeomorphism has a long history in the study of geometric topology—in the 1950s, Milnor's study of this problem led to his discovery of exotic spheres. In this talk, I will survey this classical problem and discuss its solution, which builds on the work of many people, including Wall and Stolz. This solution, obtained in joint works with Burklund, Hahn and Zhang, relies on modern techniques in the higher algebra of E_\infty-rings. Time permitting, I will also explain how to generalize this classification from highly-connected manifolds to the wider class of metastably-connected manifolds.
Abstract: The characterization of global solutions to the obstacle problem, or equivalently of null quadrature domains, is connected to the famous Shell Theorem of Netwon and has been studied for more than 90 years. In this talk I will discuss a recent result with Eberle and Weiss, where we give a conclusive answer to this question.
Abstract: The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.
Abstract:Â A qualitative stability estimate for the classical Sobolev inequality is known since the work of Bianchi and Egnell in 1991. Obtaining constructive stability estimates, i.e., giving an estimate of the stability constant, has remained an open question for a very long time. Several results of stability in strong norms have been obtained in recent years not only for the Sobolev inequality but also for the logarithmic Sobolev inequality as well as some families of Gagliardo-Nirenberg interpolation inequalities. The aim of this lecture is to provide an overview of various methods, results and open questions.
Abstract: From clinical trials to corporate strategy, randomized experiments are a reliable methodological tool for estimating causal effects. In recent years, there has been a growing interest in causal inference under interference, where treatment given to one unit can affect outcomes of other units. While the literature on interference has focused primarily on unbiased and consistent estimation, designing randomized network experiments to insure tight rates of convergence is relatively under-explored for many settings.
In this talk, we study the problem of direct effect estimation under interference. Here, the interference between experimental subjects is captured by a network and the experimenter seeks to estimate the direct effect, which is the difference between the outcomes when (i) a unit is treated and its neighbors receive control and (ii) the unit and its neighbors receive control. We present a new experimental design under which the normalized variance of a Horvitz—Thompson style estimator is bounded as $n * Var