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.