L4

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.

Topological Tverberg theory seeks r-fold analogues of classical nonembeddability results. Given a simplicial complex K, the central question is whether every continuous map from K into R^d necessarily identifies r points lying on r pairwise disjoint faces of K. The corresponding collection of faces is called a Tverberg r-partition. Perhaps surprisingly, the existence of such partitions depends on the arithmetic properties of r.

The admissible-prescribable conjecture proposes a refinement of this theory by predicting exactly which face dimensions must occur in Tverberg r-partitions. The conjecture has been verified in a number of cases, using tools such as shelling constructions and discrete Morse theory to determine the homotopy type of the relevant configuration spaces.

In this talk, we present counterexamples that settle the remaining open cases and disprove the conjecture in full generality. Our approach combines a diagrammatic description of configuration spaces with techniques from the theory of homotopy colimits of covers, allowing us to equivariantly reduce these spaces. We then show how methods from differential and PL topology, including the r-fold Whitney trick and surgery of intersections techniques, can be employed to construct the desired counterexamples.

This talk is based on forthcoming joint work with Pavle Blagojević and Florian Frick.

Fersztand, Jacquard, Nanda, and Tilmann ('24) introduced the Skyscraper Invariant, a filtration of the classical rank-invariant, for multiparameter persistence modules. It is defined by considering the Harder-Narasimhan (HN) filtration of the module along a special set of stability conditions.

This talk will begin with a post-hoc motivation for considering stability conditions on persistence modules. To compute an approximation of the Skyscraper Invariant we present a technique which, exploiting the geometry of low-dimensional bifiltrations, lets us perform a brute-force computation. We compare it against Cheng's algorithm [Cheng24] which can compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension.

To avoid unnecessary recomputation in our algorithm, we ask for which stability conditions the HN filtrations are equivalent. This partition of the space of stabililty conditions is called the wall-and-chamber structure. We show that for a finitely presented d-parameter module it is given by the lower envelopes of a set of multilinear polynomials of degree d-1. For d=2 it is then easy to compute this, enabling a faster algorithm to compute the Skyscraper Invariant up to arbitrary accuracy. As a proof of concept for data analysis, we use it to compute a filtered version of the Multiparameter Landscape for large modules from real world data.