The celebrated additive kinematic formula expresses the mean volume of the Minkowski sum of two compact convex subsets of the Euclidean space placed at random. What about non convex subsets? What about other Lie groups than the Euclidean space? In a joint work with Andreas Bernig, we prove additive kinematic formulas for compact subanalytic sets of the Euclidean space and of the 3-sphere. The key is to generalize the Minkowski sum of convex bodies by a notion of convolution of subanalytic sets introduced by Schapira in the late 80s using Euler characteristic computations. The above will of course be an excuse to discuss integral geometric formulas and constructible functions.

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.

Between 1969 and 1974, Doplicher, Haag and Roberts published a series of papers, studying the structure of the algebra of observables of general QFTs. Only very recently did those ideas get adapted to the study of discrete systems, or quantum lattice systems.

In this talk, mostly based on Corey Jones' original paper (arXiv 2304.00068), I will give an overview of the mathematical machinery behind what he called "discrete DHR theory". I will also present some of the main results that have been developed in this formalism: a new tool for the study of Quantum Cellular Automata, and a SymTFT-like construction for discrete systems.

We will begin with an overview of Stark's conjectures before discussing the case of imaginary quadratic fields, covering both the limit formula and the existence of elliptic units. The classical expositions of these are at times lacking in intuition, but thanks to Kato's deep insights 20 years ago, we can present more geometric and illuminating proofs of both results.

Fix an Artin representation rho. Work in progress by Emmanuel Lecouturier and Loïc Merel claims that the special values L(f,rho,1) for certain modular forms f see some global data related to the L-function attached to rho. We first give a brief exposition on Mazur’s Eisenstein ideal, which lies at the heart of their work. We then describe this conjectural phenomenon in a few simple cases, the last being related to a conjecture of Harris and Venkatesh.