L4

to follow

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

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.

The Euler characteristic transform (ECT) is an emerging and powerful framework within topological data analysis for quantifying the geometry of shape. The applicability of ECT has been limited due to its sensitivity to noisy data. Here, we introduce SampEuler, a novel ECT-based shape descriptor designed to achieve enhanced robustness to perturbations. We provide a theoretical analysis establishing the stability of SampEuler and validate these properties empirically through pairwise similarity analyses on a benchmark dataset and showcase it on a thymus dataset. The thymus is a primary lymphoid organ that is essential for the maturation and selection of self-tolerant T cells, and within the thymus, thymic epithelial cells are organized in complex three-dimensional architectures, yet the principles governing their formation, functional organization, and remodeling during age-related involution remain poorly understood. Addressing these questions requires robust and informative shape descriptors capable of capturing subtle architectural changes across developmental stages. We develop and apply SampEuler to a newly generated two-dimensional imaging dataset of mouse thymi spanning multiple age groups, where SampEuler outperforms both persistent homology-based methods and deep learning models in detecting subtle, localized morphological differences associated with aging. To facilitate interpretation, we develop a vectorization and visualization framework for SampEuler, which preserves rich morphological information and enables identification of structural features that distinguish thymi across age groups. Collectively, our results demonstrate that SampEuler provides a robust and interpretable approach for quantifying thymic architecture and reveals age-dependent structural changes that offer new insights into thymic organization and involution.

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.

In this talk, we give a construction of the Abelian Yang--Mills--Higgs measure on the two-dimensional torus via the variational approach initiated by Barashkov--Gubinelli. The construction is carried out through a disintegration of measures: we first construct the conditional Higgs measure given a rough gauge field, and then construct the gauge field marginal. This leads to iterated variational problems, one for the Higgs field and one for the gauge field. At the technical level, the starting point is the construction of the renormalised covariant Laplacian associated to a rough gauge field, together with the study of its resolvent. This allows us to define the covariant Gaussian free field, which serves as the reference Gaussian field for the conditional Higgs measure. Finally, we analyse the ratios of determinants that arise from the change-of-measure formula for Gaussian measures. This is joint work with Nikolay Barashkov, Ajay Chandra, Ilya Chevyrev, and Andreas Koller.