Theta operators are weight-shifting differential operators on  automorphic forms. They play an important role in studying congruences between Hecke eigenforms and their p-adic variation. For instance, the classical theta operator, which acts on q-expansions of modular forms as q·(d/dq), is used crucially in Edixhoven’s proof of the weight part of Serre’s conjecture, Katz’s construction of p-adic L-functions over CM fields, and Coleman’s classicality theorem.

Recent years have witnessed extensive works on understanding theta operators over general Shimura varieties, from both geometric and representation-theoretic perspectives. In this talk, I will hint at some aspects of this fascinating area of research. If time permits, I will discuss my ongoing work on overconvergent theta operators over Siegel Shimura varieties.

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.

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.