Erik Amezquita

The shape of things to come: plant biology meets topology

Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare, and analyze this information embedded in a robust and concise way, we turn to Topological Data Analysis (TDA), specifically the Euler Characteristic Transform (ECT). The ECT measures shape by analyzing topological features of an object at thresholds across a number of directional axes. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of a large collection of barley seeds scanned with X-ray Computed Tomography (CT) technology. We then successfully train a support vector machine (SVM) to classify 28 different accessions of barley based exclusively on the shape of their grains. While traditional shape descriptors can cluster the seeds based on their accession, topological shape descriptors can cluster them further based on their spike. We observe that combining both traditional and topological descriptors classifies barley seeds better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to comprehensively describe a multitude of “hidden” shape nuances which are otherwise not detected and offers an exciting new path to link phenotype with genotype.

Robyn Brooks

Multi-parameter Persistence and Discrete Morse Theory

Persistent Homology is a tool of Computation Topology which is used to determine the topological features of a space from a sample of data points. In this talk, I will introduce the (multi-)persistence pipeline, as well as some basic tools from Discrete Morse Theory which can be used to better understand the multi-parameter persistence module of a filtration. In particular, the addition of a discrete gradient vector field consistent with a multi-filtration allows one to exploit the information contained in the critical cells of that vector field as a means of enhancing geometrical understanding of multi-parameter persistence. I will present results from joint work with Claudia Landi, Asilata Bapat, Barbara Mahler, and Celia Hacker, in which we are able to show that the rank invariant for nD persistence modules can be computed by selecting a small number of values in the parameter space determined by the critical cells of the discrete gradient vector field. These values may be used to reconstruct the rank invariant for all other possible values in the parameter space.

Justin Curry

Exemplars of Sheaf Theory in TDA

In this talk I will present four case studies of sheaves and cosheaves in topological data analysis. The first two are examples of (co)sheaves “in the small”: (1) level set persistence—and its generalization to higher dimensions—and (2) decorated merge trees and Reeb graphs—enriched topological invariants that have enhanced classification power over traditional TDA methods. The second set of examples are focused on (co)sheaves “in the large”: (3) understanding the space of merge trees as a stratified map to the space of barcodes and (4) the organization of the persistent homology transform into a “sheaf of sheaves” on the site of constructible subsets, which provides us with a new sheaf-theoretic construction of shape space.

Mario Gomez

Curvature Sets Over Persistence Diagrams

We study an invariant of compact metric spaces inspired by the Curvature Sets defined by Gromov. The (n,k)-Persistence Set of X is the collection of k-dimensional VR persistence diagrams of any subset of X with n or less points. This research seeks to provide a cheaper persistence-like invariant for metric spaces, which we hope will aid practical computations as well as provide insights for theoretical characterizations of VR complexes. I’ll focus on theoretical results in the case n=2k+2, where we can find a geometric formula for the persistence diagram of a space with n points. We explore the application of this formula to the characterization of persistence sets of several spaces, including circles, higher dimensional spheres, and surfaces with constant curvature. We also show that persistence sets can detect the homotopy type of a certain family of graphs.

Alex McCleary

Persistence Diagrams over Finite Posets

In this talk we will explore the structure of persistence diagrams over finite posets. We will construct a notion of persistence diagram via the Mobius inversion of the birth-death function, present a stability result for these persistence diagrams, and look at the categories of persistence modules and persistence diagrams.

Samantha Moore

Hyperplane Restrictions of Indecomposable Multiparameter Persistence Modules

Understanding the structure of indecomposable multiparameter persistence modules is a difficult problem, yet is foundational for studying multipersistence. We extend a result by Buchet and Escolar to the following: If M is any finitely presented (n-1)-parameter persistence module with finite support, then there exists an indecomposable n-parameter persistence module M’ such that M is the restriction of M’ to a hyperplane. The result is a deeper understanding of how complicated the structure of multiparameter persistence modules can be.

Michael Moy

Vietoris–Rips metric thickenings: persistent homology and homotopy types

Vietoris–Rips metric thickenings are constructions similar to traditional Vietoris–Rips simplicial complexes.  In this talk, I’ll introduce metric thickenings and describe important properties that make them well-suited for studying persistent homology.  In particular, this includes my recent results showing the persistence diagrams for Vietoris–Rips metric thickenings agree with those for simplicial complexes.  I will also describe recent work to understand the homotopy types of Vietoris–Rips metric thickenings of the circle.

Sarah Percival

An efficient algorithm for the computation of Reeb graphs from roadmaps

The Reeb graph, a tool from Morse theory, has recently found use in applied topology due to its ability to track the changes in connectivity of level sets of a function. The roadmap of a set, a construction that arises in semi-algebraic geometry, is a one-dimensional set that encodes information about the connected components of a set. In this talk, I will show that the Reeb graph and, more generally, the Reeb space, of a semi-algebraic set is homeomorphic to a semi-algebraic set, which raises the algorithmic problem of computing a semi-algebraic description of the Reeb graph. We present an algorithm with singly-exponential complexity that realizes the Reeb graph of a function f: X -> Y as a semi-algebraic quotient using the roadmap of X with respect to f.

Luis Polanco

Quasi-periodicity detection with persistent cup products.

Many natural phenomena are characterized by their periodic nature, including animal locomotion, biological processes, pendulums, etc. Part of understanding periodic process is being able to differentiate them from quasiperiodic occurrences. Many advances have been made to use topological data analysis to classy and understand quasiperiodic signals. Some of these methodologies make use of persistent 2-dimensional homology to obtain quasiperiodic scores that indicate the degree of periodicity or quiasiperiodicity of a signal. There is a significant computational disadvantage in this approach since it requires, the often expensive, computation of 2-dimensional persistent homology. Our contribution uses the algebraic structure of the cohomology ring to obtain classes in the 2-dimensional persistent diagram only computing classes in dimension 1, saving valuable computational time and obtaining more reliable quasiperiodicity scores. To achieve this goal, we develop an algorithm that allow us to effectively compute the cohomological death and birth of a persistent cup product expression by taking advantage of some of the efficiencies baked in the Ripser algorithm. Finally, using these values we define a quasiperiodic score that separates periodic from quasiperiodic time series and present results for synthetic and real data sets.

Demi Qin

A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this talk, I will talk our recent work on persistence diagram hashing framework that learns a binary code representation of persistence diagrams, which allows for fast computation of distances. This framework is built upon a generative adversarial network (GAN) with a diagram distance loss function to steer the learning process. Instead of using standard representations, we hash diagrams into binary codes, which have natural advantages in large-scale tasks. The training of this model is domain-oblivious in that it can be computed purely from synthetic, randomly created diagrams. As a consequence, our proposed method is directly applicable to various datasets without the need for retraining the model. These binary codes, when compared using fast Hamming distance, better maintain topological similarity properties between datasets than other vectorized representations. To evaluate this method, we apply our framework to the problem of diagram clustering and we compare the quality and performance of our approach to the state-of-the-art.

Anna Schenfisch

Augmented Persistence Modules as Constructible Cosheaves

Recent work has reframed traditional notions of persistence modules in the language of constructible cosheaves over a stratified parameter space. However, much of this work starts from a filtration rather than from the computed output of a persistence algorithm, which generally has omitted information about `instantaneous events.’ By not omitting this data, we arrive at an augmented persistence module, which also is naturally a constructible cosheaf. In this talk, we explore the idea of persistence modules as cosheaves and the connection between general persistence module cosheaves and their augmented counterparts. This setting then allows for further study of persistence modules, including K-theoretic results.

Sarah Tymochko

Topological Signal Processing using Complex Networks

While topological tools have been used to study networks from numerous applications, the use of these tools to study time series data is fairly new. Ordinal partition networks are a type of graph representation of a time series that can capture features indicative of periodic or chaotic behavior. In our work, we study modifications to the ordinal partition network method to determine if including additional features such as directionality captures more information about the underlying system than the original method.

Ling Zhou

Persistent cup-length

In topological data analysis, one uses persistent homology and its dual notion persistent cohomology to study the evolution of (co)homology across a filtration. Compared with homology, cohomology is enriched with a graded ring structure given by the cup product operation. In this talk, we utilize the cup product operation to define a new invariant for the persistent cohomology ring, called the persistent cup-length function, which is able to extract and encode additional information across a filtration, compared to the persistent (co)homology vector space.

The persistent cup-length function is a lifted version of the standard invariant: the cup-length of a cohomology ring, which is the maximum number of cocycles having non-zero cup product. We show that the persistent cup-length function is stable under suitable interleaving-type distances, and we devise a polynomial time algorithm for its computation.

Lori Ziegelmeier

Obtaining and Optimizing Persistent Homology Cycle Representatives

Abstract: Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, matrices in this domain are typically large, with rows and columns numbered in billions, which makes obtaining these cycle representatives challenging. Low-rank approximation of such arrays typically destroys essential information; thus, new mathematical and computational paradigms are needed for very large, sparse matrices. We introduce the framework of a basis matching complex and present the U-match matrix factorization scheme to address this challenge. The language of basis matching complexes allows us to extend the capabilities of persistence solvers to include a variety of sought-after features, including computation of cycle representatives for persistent homology classes. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. We provide a study of the effectiveness and computational cost of several $\ell_1$-minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms.