The Geometric Realization of AATRN (Applied Algebraic Topology Research Network)

I will use tools from applied algebraic topology for feature selection on time series data. I will present a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding point cloud. Our approach produces a piecewise-linear Lipschitz gradient path in the standard geometric simplex that starts at the barycenter, which weights the variables equally, and ends at the score. Adding Gaussian perturbations to the input data and taking expectations results in a mean gradient path that satisfies a strong law of large numbers and central limit theorem. Our theory is motivated by the analysis of the neuronal activities of the nematode C. elegans, and our method selects an informative subset of the neurons that optimizes the coordinated dynamics.

This is joint work with Johnathan Bush (James Madison University).

Machine learning, especially the use of neural netowrks, have shown great success in a broad range of applications. In recent years, we have also seen significant advancement in effective neural architectures for learning on more complex data, such as point sets (note that here, each input sample is a set of points) or graphs. Examples include DeepSet, Transformer and Sumformer. This facilitates the development of neural network based approaches to solve (potentially hard) geometric optimization problems, such as the minimum enclosing ball of an input set of points, in a data-driven manner. In this talk, I will describe some of our recent exploration in designing effective and efficient neural models for several geometric problems: (1) estimating the Wasserstein distance between two input point sets, and (2) a family of shape fitting problems (e.g, fitting minimal enclosing balls). Our goal is to have a neural network model of bounded size (independent to input size) that can approximately solve a target problem for input of arbitrary sizes. This talk is based on joint work with S. Chen, T. sidiropoulos and O. Ciolli.

We introduce a new class of graph kernels based on tropical geometry and the topology of metric graphs. Unlike traditional graph kernels that are defined by graph combinatorics (nodes, edges, subgraphs), our approach considers only the geometry and topology of the underlying metric space. A key property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs that represent different underlying spaces. Our kernels are efficient to compute and depend only on the graph genus rather than the size. In label-free settings, our kernels outperforms existing methods, which we showcase on synthetic, benchmarking, and real-world data and real-world road data. Joint work with Yueqi Cao.

We establish a geometric construction of Kashiwara crystals on the irreducible components of the varieties of multiparameter persistence modules. Our approach differs from the seminal work of Kashiwara and Saito, as well as subsequent related works, by emphasizing commutative relations rather than preprojective relations for a given quiver. Furthermore, we provide explicit descriptions of the Kashiwara operators in the fundamental cases of 1- and 2-parameter persistence modules, offering concrete insights into the crystal structure in these settings.

This talk introduces Möbius homology, a new homological invariant for modules over finite posets that categorifies the classical Möbius inversion formula. Given a module over a poset (a functor from a poset to an abelian category), we construct its Möbius homology by localizing a simplicial cosheaf over the order complex of the poset. Our main result shows that the Euler characteristic of Möbius homology recovers the Möbius inversion, providing a topological lift of this fundamental combinatorial operation.

The primary motivation for developing this theory comes from persistent homology. We demonstrate how Möbius homology provides a natural categorification of persistence diagrams for modules over arbitrary finite posets, extending beyond the classical totally ordered case.

Many processes in the life sciences are inherently multi-scale and dynamic. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) provides a multi-scale summary of data. We review persistent homology and then highlight applications of single parameter persistent homology and multiparameter persistence to proteins, cancer and spatial transcriptomics. Time permitting, we present in-progress work for quantifying spatio-temporal trajectories, which builds on work by Kim and Memoli, Bubenick, Vipond and Lesnick, Bender and Gäfvert.

Many tasks in data science, including distributed modeling, dimensionality reduction, data alignment and coordinatization, can be phrased as local-to-global inference problems with well-defined topological obstructions. The goal of this talk is to describe several of these constructions, their applications to specific machine learning problems, and the emergent theoretical/algorithmic challenges in geometric and topological data analysis.

We study the geometric decomposition of the (reduced) Persistent Homology Transform (PHT). We focus on two special classes of objects, star shapes and cutouts. We prove that the PHT_0 of the former can be segmented into smaller, simpler regions known as "sectors", and that the PHT of the latter can be decomposed into vines coming from simpler objects. These results provide a solid basis for future implementations of vines' decomposition of PHT, as they considerably reduce the complexity of the problem.

Morse theory is a powerful mathematical framework for studying the topology of manifolds through the analysis of critical points of smooth real-valued functions. By linking differential analysis with topology, it reveals how local function behavior affects global topological structure. Over the past two decades, Morse theory has played a central role in major developments in random topology, largely due to its inherently local-to-global perspective. In this talk, we will review key results in this area, with a focus on the pivotal contributions of Morse theory. We will also outline some of the main open challenges in the probabilistic Morse-theoretic analysis.

The interleaving distance is a fundamental metric in Topological Data Analysis (TDA), with applications ranging from Reeb graphs and persistence modules to more general categorical representations of data. This framework assesses the similarity of two structures by finding a pair of parameterized maps — called an interleaving — that give a notion of an approximate isomorphism; the minimum parameter for which such an interleaving exists quantifies their distance. While the interleaving distance for 1-parameter persistence modules can be efficiently computed in polynomial time, many other cases are NP-complete — including the setting we focus on here: mapper graphs and Reeb graphs. In this talk, we introduce a new computational view of the interleaving distance for mapper graphs. We propose a loss function that measures how far a candidate structure falls short of a true interleaving. By framing the search for an interleaving as an integer linear program, we can utilize ILP solvers which apply heuristics but are often able to compute an exact answer. While the nature of ILPs mean that these results do not come with global guarantees, we show that in many cases they successfully identify the true interleaving distance. We will also demonstrate how this framework can aid in shape comparison and be naturally generalized to other TDA structures of interest.

The category of pseudotopological spaces is the closed cartesian hull of topological spaces, and contains within it the categories of topological spaces, reflexive graphs, and metric spaces endowed with a privileged scale. We show how many classical topological invariants can be generalized to pseudotopological spaces, and how this category and others like it simplify known results and lead to new ones.

Vineyards are a common way to study persistence diagrams of a data set which is changing, as strong stability means that it is possible to pair points in ``nearby'' persistence diagrams, yielding a family of point sets which connect into curves when stacked. Recent work has also studied monodromy in the persistent homology transform, demonstrating some interesting connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in ℝ2. In this work, we re-characterize monodromy in terms of periodicity of the associated vineyard of persistence diagrams. We construct a family of objects in any dimension which have non-trivial monodromy for l-persistence of any periodicity and for any l. More generally we prove that any knot or link can appear as a vineyard for a shape in ℝd, with d≥3. This shows an intriguing and, to the best of our knowledge, previously unknown connection between knots and persistence vineyards. In particular this shows that vineyards are topologically as rich as one could possibly hope.

In this talk I will introduce CocycleHunter, a pipeline for identifying and analyzing circular structure in gene expression data, which integrates methods from topological data analysis with geometric lead-lag analysis. Our method provides a powerful, cohomology-based technique for estimating the phase of genes exhibiting cyclic expression patterns (gene cascades), which has been validated on synthetic RNA transcription models, as well as on real datasets.  I’ll explain the math behind the pipeline and illustrate its application to gene expression data.

Topology interfaces with data science primarily in two ways: by providing robust, deformation-invariant descriptors that quantify theshape and structure of data, and by enabling visualizations of high-dimensional data that preserve topological features. Theseapproaches, grounded in algebraic and combinatorial topology, yield practical tools for data analysis when efficient implemention isprovided.In this talk, I will survey topological methods for analyzing multi-valued data, including constructions based on the Euler characteristic and techniques inspired by phylogenetics. I will also discuss recent developments in applying topological summaries to statistical hypothesis testing, particularly for nonparametric goodness-of-fit testing in high dimensions. Finally, I will explore advanced visualization strategies based on Mapper and its generalizations, which provide interpretable representations of complex data sets. The presentation will focus on both the theoretical foundations and applied outcomes of these techniques.

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.

We introduce a combinatorial condition on simplicial complexes which implies n-connectivity. We use this to study random simplicial complexes. This is part of a joint work with Michael Farber