Persistent homology is an algebraic tool for quantifying topological features of shapes and functions, which has recently found wide applications in data and shape analysis. In the first and introductory part of this talk I recall the underlying ideas and basic concepts of this very active field of research. In the second part, I plan to sketch a concrete application of this concept to digital image processing.

# Past Applied Topology Seminar

This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?

This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.

How can we adapt the Topological Data Analysis (TDA) pipeline to use several filter functions at the same time? Two orthogonal approaches can be considered: (1) running the standard 1-parameter pipeline and doing statistics on the resulting barcodes; (2) running a multi-parameter version of the pipeline, still to be defined. In this talk I will present two recent contributions, one for each approach. The first contribution considers intrinsic compact metric spaces and shows that the so-called Persistent Homology Transform (PHT) is injective over a dense subset of those. When specialized to metric graphs, our analysis yields a stronger result, namely that the PHT is injective over a subset of full measurem which allows for sufficient statistics. The second contribution investigates the bi-parameter version of the TDA pipeline and shows a decomposition result "à la Crawley-Boevey" for a subcategory of the 2-parameter persistence modules called "exact modules". This result has an impact on the study of interlevel-sets persistence and on that of sheaves of vector spaces on the real line.

This is joint work with Elchanan Solomon on the one hand, with Jérémy Cochoy on the other hand.

Steady state chemical reaction models can be thought of as algebraic varieties whose properties are determined by the network structure. In experimental set-ups we often encounter the problem of noisy data points for which we want to find the corresponding steady state predicted by the model. Depending on the network there may be many such points and the number of which is given by the euclidean distance degree (ED degree). In this talk I show how certain properties of networks relate to the ED degree and how the runtime of numerical algebraic geometry computations scales with the ED degree.

Single parameter persistent homology has proven to be a useful data analytic tool and single parameter persistence modules enjoy a concise description as a barcode, a complete invariant. [Bubenik, 2012] derived a topological summary closely related to the barcode called the persistence landscape which is amenable to statistical analysis and machine learning techniques.

The theory of multidimensional persistence modules is presented in [Carlsson and Zomorodian, 2009] and unlike the single parameter case where one may associate a barcode to a module, there is not an analogous complete discrete invariant in the multiparameter setting. We propose an incomplete invariant derived from the rank invariant associated to a multiparameter persistence module, which generalises the single parameter persistence landscape in [Bubenik, 2012] and satisfies similar stability properties with respect to the interleaving distance. Our invariant naturally lies in a Banach Space and so is naturally endowed with a distance function, it is also well suited to statistical analysis since there is a uniquely defined mean associated to multiple landscapes. We shall present computational examples in the 2-parameter case using the RIVET software presented in [Lesnick and Wright, 2015].

Topological data analysis (TDA) is a robust field of mathematical data science specializing in complex, noisy, and high-dimensional data. While the elements of modern TDA have existed since the mid-1980’s, applications over the past decade have seen a dramatic increase in systems analysis, engineering, medicine, and the sciences. Two of the primary challenges in this field regard modeling and computation: what do topological features mean, and are they computable? While these questions remain open for some of the simplest structures considered in TDA — homological persistence modules and their indecomposable submodules — in the past two decades researchers have made great progress in algorithms, modeling, and mathematical foundations through diverse connections with other fields of mathematics. This talk will give a first perspective on the idea of matroid theory as a framework for unifying and relating some of these seemingly disparate connections (e.g. with quiver theory, classification, and algebraic stability), and some questions that the fields of matroid theory and TDA may mutually pose to one another. No expertise in homological persistence or general matroid theory will be assumed, though prior exposure to the definition of a matroid and/or persistence module may be helpful.

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. I will explain how we can use tropical-like functions to coordinatize the space of persistence barcodes. These coordinates are stable with respect to the bottleneck and Wasserstein distances. I will also show how they can be used in practice.

Phase transitions and critical phenomena are ubiquitous in Nature. They permeate physics, chemistry, biology and complex systems in general, and are characterized by the role of correlations and fluctuations of many degrees of freedom. From a mathematical viewpoint, in the vicinity of a critical point, thermodynamic quantities exhibit singularities and scaling properties. Theoretical attempts to describe classical phase transitions using tools from differential topology and Morse theory provided strong arguments pointing that a phase transition may emerge as a consequence of topological changes in the configuration space around the critical point.

On the other hand, much work was done concerning the topology of networks which spontaneously emerge in complex systems, as is the case of the genome, brain, and social networks, most of these built intrinsically based on measurements of the correlations among the constituents of the system.

We aim to transpose the topological methodology previously applied in n-dimensional manifolds, to describe phenomena that emerge from correlations in a complex system, in which case Hamiltonian models are hard to invoke. The main idea is to embed the network onto an n-dimensional manifold and to study the equivalent to level sets of the network according to a filtration parameter, which can be the probability for a random graph or even correlations from fMRI measurements as height function in the context of Morse theory. By doing so, we were able to find topological phase transitions either in random networks and fMRI brain networks. Moreover, we could identify high-dimensional structures, in corroboration with the recent finding from the blue brain project, where neurons could form structures up to eleven dimensions.The efficiency and generality of our methodology are illustrated for a random graph, where its Euler characteristic can be computed analytically, and for brain networks available in the human connectome project. Our results give strong arguments that the Euler characteristic, together with the distributions of the high dimensional cliques have potential use as topological biomarkers to classify brain Networks. The above ideas may pave the way to describe topological phase transitions in complex systems emerging from correlation data.

I will present Vidit Nanda's paper "Local homology and stratification" (https://arxiv.org/abs/1707.00354), and briefly explain how in my master thesis I am applying ideas from the paper to study word embedding problems.

Abstract of the paper: We outline an algorithm to recover the canonical (or, coarsest) stratification of a given regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a finite sequence of categories whose colimit recovers the canonical strata via (isomorphism classes of) its objects. The entire process is amenable to efficient distributed computation.

This is the continuation of a discussion of Ezra Miller's paper https://arxiv.org/abs/1709.08155.