L4

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.

TBA

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.
 

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