Abstract: In a recent paper with Mark Kisin and Yihang Zhu, we proved the stable trace formula for Shimura varieties of abelian type. (This was the subject of Zhu’s talks in early October.) We will discuss applications of this formula. After a broad introduction to such applications, we will specialize to the problem of describing the cohomology of Shimura varieties (joint work with Kisin and Zhu).
Abstract: A celebrated result by Donaldson asserts that the space of almost-Fuchsian manifolds admits a natural hyperKähler structure invariant under the action of the mapping class group, extending the Weil-Petersson Kähler structure of Teichmüller space. In this talk we will discuss the occurrence of a similar phenomenon for the deformation space of globally hyperbolic Anti-de Sitter 3-manifolds. In particular we will see how such space carries a para-hyperKähler structure, where a pseudo-Riemannian metric and 3 symplectic structures coexist with a integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. This project is a joint work with Andrea Seppi (Université Grenoble Alpes) and Andrea Tamburelli (Rice University).
Abstract: Since work of Gromov and Lawson around 1980, we have known (under favorable circumstances) necessary and sufficient conditions for a closed manifold to admit a Riemannian metric of positive scalar curvature, but not much was known about analogous results for manifolds with boundary (and suitable boundary conditions). In joint work with Shmuel Weinberger of the University of Chicago, we give necessary and sufficient conditions in many cases for compact manifolds with non-empty boundary to admit: (a) a positive scalar curvature metric which is a product metric in a neighborhood of the boundary or (b) a positive scalar curvature metric with positive mean curvature on the boundary.
Abstract: We will discuss multiple CUR algorithms. The focus will be on the algorithms and how they perform in practice. Algorithms include deterministic and randomized methods and will be presented in chronological order. Applications may also be presented.
Abstract: As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm's behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast.
Abstract: One classical way to study rational or integral solutions of apolynomial equation is to look at the simpler questions of the various congruence equations modulo n for all integers n. The conjecture of Birch and Swinnerton-Dyer predicts that, for elliptic curves (defined by equations of the form y^2=x^3+ax+b with integer coefficients), the data from these congruence equations mod n should actually encode much information on the solutions in rational numbers. In the first talk we will discuss a generalization of the question to the product of several elliptic curves, where, instead of rational points, we look for algebraic cycles (i.e., parameter solutions) modulo suitable equivalence relations (rational equivalence, Abel—Jacobi or its p-adic version). In particular, I'll report some recent results on a conjecture of Bloch-Kato.
Abstract: A common task in many data-driven applications is to find a low dimensional manifold that describes the data accurately. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there is no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise. In this talk, we will present a method that estimates a manifold and its tangent in the ambient space. Moreover, we establish rigorous convergence rates, which are essentially as good as existing convergence rates for function estimation.
1) Iwaki, K. and Nakanishi, T. (2014). Exact WKB analysis and cluster algebras. Journalof Physics A: Mathematical and Theoretical, 47(47), 474009. 2) Kawai, T. and Takei, Y. (2005). Algebraic analysis of singular perturbation theory. American Mathematical Society. 3) Voros symbols as cluster coordinates, Journal of Topology, 12(2019), 1031–1068.
Abstract: We study the problem of decision-making in the setting of a scarcity of shared resources when the preferences of agents are unknown a priori and must be learned from data. Taking the two-sided matching market as a running example, we focus on the decentralized setting, where agents do not share their learned preferences with a central authority. Our approach is based on the representation of preferences in a reproducing kernel Hilbert space, and a learning algorithm for preferences that accounts for uncertainty due to the competition among the agents in the market. Under regularity conditions, we show that our estimator of preferences converges at a minimax optimal rate. Given this result, we derive optimal strategies that maximize agents' expected payoffs and we calibrate the uncertain state by taking opportunity costs into account. We also derive an incentive-compatibility property and show that the outcome from the learned strategies has a stability property. Finally, we prove a fairness property that asserts that there exists no justified envy according to the learned strategies. This is a joint work with Michael I. Jordan.