Hannah Alpert

Homology of configuration spaces of squares in a rectangle

We consider the configuration space of n unit squares sliding in a p by q rectangle.  In which degrees is its homology concentrated?  Squares in a rectangle serve as a model for molecules in a container.  Can we detect (approximately) whether the substance is a solid, liquid, or gas, using only the topology of the configuration space?  Even very basic questions about these configuration spaces tend to be unresolved, so there are many appealing directions for future research.

Hubert Wagner

Proofs, refutations… and topological computations

Shreya Arya

A Sheaf Theoretic Construction of Shape Space

In this talk I will present a sheaf theoretic construction of shape space (or the space of all shapes). We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to “glue” PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision. This is joint work with Justin Curry and Sayan Mukherjee.

Robin Belton

Extremal Event Graphs: A (Stable) Tool for Analyzing Noisy Time Series

Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise $L_\infty$ distances between functions in the time series data. Lastly, we provide implementations of these methods to answer questions concerning circadian rhythms and gene regulatory networks.

Jonathan Treviño-Marroquín

Limit spaces, looking for a universal covering

Limit spaces is a generalization of topology where we indicate what filters converge under some axioms. In this article, we introduce covering maps and set forth some necessary conditions for a construction for a universal covering space.

Sarah McGuire

Learned Simplicial Complex Coarsening

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and minimizing overfitting. In this talk, I will discuss a pooling layer for data structured as simplicial complexes, which are generalizations of graphs that include of higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. We propose a simplicial coarsening scheme built upon soft partitions of vertices, which allows us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. The simplicial pooling method builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice, the pooling operations are computed via a series of matrix operations, its topological motivation is based on unions of stars of simplices and the nerve complex.

Zhengchao Wan

A generalization of the persistent Laplacian to simplicial maps

The Graph Laplacian is a fundamental object in graph analysis and optimization, playing a key role in a wide range of applications. By adopting a topological viewpoint, this operator can be extended to simplicial complexes, providing a means to perform signal processing on (co)chains of simplicial complexes. In recent years, the concept of Persistent Laplacian has been introduced and studied for a pair of simplicial complexes related by an inclusion relation, which broadens the use of Laplace-based operators in this context. 

Our recent work has gone one step further, successfully expanding the scope of the Persistent Laplacian to handle simplicial complexes connected by simplicial maps. Our construction is non-trivial and this setting of simplicial maps arises frequently in practice, for example when relating a coarsened simplicial representation with an original representation. In this talk, I will provide an overview of the Persistent Laplacian and then introduce our new construction and highlight its properties, including its ability to recover persistent Betti numbers.

Edgar Jaramillo Rodriguez 

Combinatorial Methods for Barcode Analysis

A barcode is a finite multiset of intervals on the real line. Barcodes are important objects in topological data analysis (TDA), where they serve as summaries of the persistent homology groups of a filtration. We introduce a new combinatorial invariant associated to barcodes by mapping each barcode to a multipermutation, i.e., a permutation of some multiset, which captures the overlapping arrangement of its bars. We call the set all such multipermutations the space of combinatorial barcodes. We then define an order on this space and show that the resulting poset is a graded lattice. Finally, we show that cover relations in this lattice can also be used to determine the set of barcode bases of persistence modules. If time allows, we will also discuss a generalization of this construction and its applications to TDA. 

Abhishek Rathod

Approximation of the multicover bifiltration

Abstract: For a finite point set P \subset R^d, let M_{r,k} denote the set of points in R^d that are within distance r of at least k points in P. Allowing r and k to vary yields a 2-parameter family of spaces M, called the multicover bifiltration of P. It is a density-sensitive extension of the union-of-balls filtration commonly considered in TDA. It is robust to outliers in a strong sense, which motivates the problem of efficiently computing it (up to homotopy). A recent algorithm of Edelsbrunner and Osang computes a polyhedral model of M called the rhomboid bifiltration. In this work, we introduce an approximation of M (up to homotopy) which extends a version of the rhomboid tiling and devise an efficient algorithm to compute it.

The talk is based on joint work with Uli Bauer, Tamal Dey and Michael Lesnick.

Tananun Songdechakraiwut

Fast Topological Clustering with Wasserstein Distance

The topological patterns exhibited by many real-world networks motivate the development of topology-based methods for assessing the similarity of networks. However, extracting topological structure is difficult, especially for large and dense networks whose node degrees range over multiple orders of magnitude. In this talk, I will present a novel and computationally practical topological clustering method that clusters complex networks with intricate topology using principled theory from persistent homology and optimal transport. Such networks are aggregated into clusters through a centroid-based clustering strategy based on both their topological and geometric structure, preserving correspondence between nodes in different networks. The notions of topological proximity and centroid are characterized using a novel and efficient approach to computation of the Wasserstein distance and barycenter for persistence barcodes associated with connected components and cycles. The proposed method is demonstrated to be effective using both simulated networks and measured functional brain networks. This talk abstract is adapted from the corresponding article published at the International Conference on Learning Representations (ICLR 2022).

Hans Riess 

The Tarski Laplacian and Beyond

Ordered sets with binary meet and join operations, known as lattices, have emerged in economics as key mathematical objects for modeling preferences, choice, knowledge, belief, and more. By defining an appropriate Laplacian, the dynamics of heat flow can be applied to these settings. The Tarski Laplacian is a lattice-theoretic analogue of the graph (connection) Laplacian, the key object from the corpus of signal processing on irregular domains. In the Hodge-Tarski Theorem (Ghrist & Riess, 2022), fixed points of the Tarski Laplacian correspond to global sections (consistent assignments of data) of the underlying lattice-valued sheaf. Our theory marks an important first step towards the development of a combinatorial Hodge Theory for cellular sheaves valued in non-abelian categories. On the side of applications, heat flow dynamics is but one of many regimes for, e.g., preference change in the multi-agent setting.

Abigale Hickok

Applications of TDA to Spatial Data

In this talk, we will explore applications of persistent homology to geospatial and geospatiotemporal data. We will study the geographic distribution of polling sites as well as two COVID-19 data sets (LA infection rates and NYC vaccination rates). For the polling site data, we use persistent homology to identify “holes in coverage”–geographic areas where people cannot easily access a polling place in order to vote in an election. For the COVID-19 data, we use persistent homology to identify “anomalies” (local maxima in the infection-rate data and local minima in the vaccination data) and analyze global spatial structure; we use vineyards to study how the data changes with time. Each data set requires a unique filtered-complex construction in order to model the problem appropriately.

Baihan Lin

The Topology and Geometry of Neural Representations.

A central question for neuroscience is how to characterize brain representations of perceptual and cognitive content. An ideal characterization should distinguish different functional regions while abstracting from idiosyncrasies of individual brains that do not correspond to computational differences. Previous studies have characterized brain representations by their representational geometry, which is defined by the representational dissimilarity matrix (RDM), a summary statistic that abstracts from the roles of individual neurons (or responses channels) and defines the discriminability of stimulus dichotomies. Here we explore a further step of abstraction: from the geometry to the topology of brain representations. We propose a family of geo-topological summary statistics that generalizes the RDM to characterize the topology while deemphasizing the geometry. We evaluate this family of statistics in terms of the sensitivity and specificity for model selection using both functional MRI (fMRI) data and simulations. In the simulations, the ground truth is a data-generating layer representation in a neural network model and the models are different layers in other model instances (trained from different random seeds). In fMRI, the ground truth is a visual area and the models are the same and other areas measured in different subjects. Results demonstrate that geo-topological summary statistics can help reveal the computational signatures of neural network layers and brain regions.