Applied Topology Seminar

In this talk, we present a topological framework for interpreting the latent representations of Multilayer Perceptrons (MLPs) [1] using tools from Topological Data Analysis. Our approach constructs a simplicial tower, a sequence of simplicial complexes linked by simplicial maps, to capture how the topology of data evolves across network layers. This construction is based on the pullback of a cover tower on the output layer and is inspired by the Multiscale Mapper algorithm. The resulting commutative diagram enables a dual analysis: layer persistence, which tracks topological features within individual layers, and MLP persistence, which monitors how these features transform across layers. Through experiments on both synthetic and real-world medical datasets, we demonstrate how this method reveals critical topological transitions, identifies redundant layers, and provides interpretable insights into the internal organization of neural networks.

An arrangement of hypersurfaces in projective space is strict normal crossing if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

Joint work with T. Kahle, B. Sturmfels, M. Wiesmann

The cover of a dataset is a fundamental concept in computational geometry and topology. In TDA (topological data analysis), it is especially used in computing persistent homology and data visualisation using Mapper. However only rudimentary methods have been used to compute a cover. In this talk, we formulate the cover computation problem as a general optimisation problem with a well-defined loss function, and use gradient descent to solve it. The resulting algorithm, ShapeDiscover, substantially improves quality of topological inference and data visualisation. We also show some preliminary applications in scRNA-seq transcriptomics and the topology of grid cells in the rats' brain. This is a joint work with Luis Scoccola and Heather Harrington.

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

The Morse-Conley complex is a central object in information compression in topological data analysis, as well as the application of homological algebra to analysing dynamical systems. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading.  In this work, we give a purely algebraic derivation of the Morse-Conley complex using homological perturbation theory. Unlike Forman’s discrete Morse theory for cellular complexes, our algebraic formulation does not require the computation of acyclic partial matchings of cells.  We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex.  This talk features joint work with Álvaro Torras Casas and Ulrich Pennig in "Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory" (arXiv:2503.09301). 
 

Multiparameter persistent homology is a generalization of persistent homology that allows for more than a single filtration function. Such constructions arise naturally when considering data with outliers or variations in density, time-varying data, or functional data. Even though its algebraic roots are substantially more complicated, several new invariants have been proposed recently. In this talk, I will go over such invariants, as well as their stability, vectorizations and implementations in statistical machine learning.

In this talk, I will discuss Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. This framework allows for the comparison of measures with different total masses by introducing an auxiliary set that compensates for mass discrepancies. Within this setting, classical characterisations of optimal transport plans extend naturally, and the resulting spaces of measures are shown to be complete, separable, geodesic, and non-branching, provided the underlying space possesses these properties. Moreover, we prove that the spaces of measures 
equipped with the $L^2$-optimal partial transport metric inherit non-negative curvature in the sense of Alexandrov. Finally, generalised spaces of persistence diagrams embed naturally into these spaces of measures, leading to a unified perspective from which several known geometric properties of generalised persistence diagram spaces follow. These results build on recent work by Divol and Lacombe and generalise classical results in optimal transport.