L4

I will describe a circle-valued Morse theory for simplicial complexes. The central objects of study are partial matchings which admit certain zigzag cycles; these cyclic matchings lift canonically to acyclic matchings on the infinite cyclic cover of the underlying simplicial complex. From the lifted acyclic matchings, we obtain a finitely generated Morse chain complex defined over the Novikov ring, which consists of power series in one variable with finite negative support. We then establish a quasi-isomorphism between this Morse-Novikov complex and the simplicial chain complex of the cyclic cover, duly completed over the Novikov ring. As a pleasant consequence, we can define new computable invariants to detect (obstructions to) the fiberedness of tame knots.

In this talk we summary some recent progress on limit theorems for the Wasserstein distance of empirical measures of Markov processes. For symmetric diffusion processes on Riemannian manifold possibly with reflecting or killing boundary, the sharp convergence rate is derived with renormalization limit formulated by using the spectrum of the generator. Moreover, a general framework is established to estimate the convergence rate in Wasserstein distance of empirical measures for ergodic Markov processes.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

I will discuss stochastic systems of interacting particles with non-overlapping constraints, which give rise to so-called excluded-volume interactions. The aim is to derive effective macroscopic equations governing the evolution of particle densities from the underlying microscopic dynamics. When particles possess nontrivial size or shape, geometric constraints become essential: they complicate the coarse-graining process and strongly influence the emergent behaviour of the system. I will present two representative examples, hard spheres and infinitely thin needles, highlighting how geometry enters the macroscopic description

The global Langlands-Kottwitz method seeks to express Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of (twisted) orbital integrals; the latter are central objects in local harmonic analysis which enter the Arthur-Selberg trace formula. While this method is well studied, we present a new local analogue: a formula relating the cohomology of local Shimura varieties to twisted orbital integrals. This local formula bridges the point-counting formula for global Shimura varieties with the point-counting formula for Igusa varieties. As an application of our local formula, we propose a new approach, based on categorical Langlands, towards Rapoport's vanishing conjecture on certain twisted orbital integrals. This conjecture is itself a key ingredient in the global Langlands-Kottwitz method for a non-quasi-split prime. This is joint work with Rong Zhou.

I will report on a joint work in progress with Egor Morozov proving that for generic elements in several families of Laplace-type operators invariant under a finite group action, all eigenspaces are irreducible representations. In particular, for the case of Laplace-Beltrami operators, this provides a natural generalization of Uhlenbeck's result on the generic simplicity of the spectrum to the equivariant setting. Moreover, this extends previous work of Zelditch and solves the finite group case of a well-known question raised by Guillemin and Yau. For Schrödinger operators, our results rigorously underpin the notion of accidental degeneracy for certain quantum-mechanical systems with finite symmetry. Our approach involves modern methods of equivariant transversality which we extend to higher dimensions.