The seminar will take place online until further notice. If you wish to participate please contact Adélie Garin , Celia Hacker or Kathryn Hess to receive the zoom link.

### Program

### Abstracts

**D. Méndez: ** Persistent homology is one of the most successful tools in Topological Data Analysis, having been applied in numerous scientific domains such as medicine, neuroscience, robotics, and many others. However, a fundamental limitation of persistent homology is its inability to incorporate directionality.

In this talk we will introduce a theory of persistent homology for directed simplicial complexes which detects directed cycles in odd dimensions. To do so, we introduce a homology theory with coefficients in semirings for these complexes: by explicitly removing additive inverses, we can detect directed cycles algebraically. We will also exhibit some of the features of this persistent homology theory, including its stability and how the obtained persistent diagrams relate to those obtained from persistent homology with ring coefficients. We will end the talk by highlighting some of the computational challenges towards the effective computation of the directed persistent diagram of a pointcloud.

**S. Chowdhury: ** Geometric and topological data analysis methods are increasingly being used in human neuroimaging studies to derive insights into neurobiology and behavior. We will begin by describing a pipeline that utilizes the Mapper algorithm to produce network representations of whole-brain activity during ongoing cognition. When applying this pipeline at scale across clinical populations, however, generating consistent insights requires the development of statistical learning techniques such as averaging and PCA across graphs without known node correspondences. We formulate this problem using the Gromov-Wasserstein (GW) distance and present a recently-developed Riemannian framework for GW-based graph averaging, partitioning, and tangent PCA. This framework permits using derived network representations beyond graph geodesic distances or adjacency matrices. In particular, we show that compared to state-of-the-art implementations that use adjacency matrix formulations, a spectral network representation leads to improved accuracy and runtime in graph learning tasks. Additionally, we observe that the spectral approach to GW graph partitioning corresponds to a generalization of Fiedler bipartitioning, thus suggesting new avenues for rigorous analysis of the GW problem.

** M. Catanzaro: ** In this talk, we will explore stochastic motion of subcomplexes inside CW complexes. We refer to this as a Markov CW-chain and it serves as a generalization of a random walks on a graph, and a discretization of stochastic flows on smooth manifolds.

We will define a notion of stochastic current, connect it to classical electric current, and show it satisfies quantization. Along the way, we will define the main combinatorial objects of study, namely spanning trees and spanning co-trees in higher dimensions,and relate these to the dynamics.

**R. Tinarrage: **Persistent homology can be seen as an answer to the following estimation problem: given a finite sample of a nice subset of the Euclidean space, estimate the homology groups of the nice subset. By nice subset we mean a C^2-submanifold, or more generally a positive-reach subset, and by homology groups we mean singular homology groups over a finite field. However, in algebraic topology, there exists many other topological invariants than homology groups. In this talk, we will deal with Stiefel-Whitney classes. These classes are associated to any topological space endowed with a vector bundle structure, and they carry more topological information than the cohomology groups alone. Our new estimation problem is the following: given a finite sample of a vector bundle, estimate the Stiefel-Whitney classes of the vector bundle. We will adopt a persistent approach.

**B. Wang: **We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f: X → R. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of (X,f) approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. We introduce a variant of the classic mapper graph of Singh et al. (2007), referred to as the enhanced mapper graph. We show that the enhanced mapper graph reduces the information loss during summarization and may be of independent interest. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data. This is joint work with Adam Brown,·Omer Bobrowski, and Elizabeth Munch.

https://arxiv.org/abs/1909.03488

**C. Hacker: ** Given the success of single-parameter persistence in data analysis and the fact that some systems warrant analysis across multiple parameters, it is highly desirable to develop data analysis pipelines based on multi-parameter persistence in order to study multi-variate data. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter.

In this talk, we will discuss how discrete Morse theory may be used to compute the rank invariant for a multi-parameter persistence module. We will show that the rank invariant of a persistence module is completely determined by its values at points whose coordinates are critical, with respect to a discrete Morse function. These critical points partition the set of all lines in the parameter space into equivalence classes, such that the rank invariant along lines in the same class are also equivalent. We will show that we can deduce all persistence diagrams of rank invariants of a given class from the persistence diagram of a representative rank invariant in that class. This is joint work with the WinCompTop group lead by Claudia Landi.

**N. Berkouk: ** The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties. Indeed, it has been shown that there exists no complete discrete invariant for persistence modules with many parameters, such as so-called barcodes in the one-parameter case. To tackle this problem, some new algebraic invariants have been proposed to study multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of the stability of these invariants has raised few attention so far, and many of the proposed invariants do not satisfy a naive form of stability. In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance, hence opening the door to performing homological algebra operations while “keeping track” of stability. This approach, we believe, can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in the case with more than one parameter).

**M.K. Youssef: ** Interactions are ubiquitous; our quest of understanding how viruses spread within a population, how information circulates on social media, or how signaling pathways are regulated within a living cell made it strikingly clear that these systems are poorly explained when focusing on the constituents that make up these systems, in isolation. Instead, placing the spotlight on the interactions between these constituents naturally leads to the field of network science, where networks provide a unifying framework for studying collective properties that emerge from interactions: scale-free degree distributions give mechanistic insights into the spread of information (or viruses) while assortativity measures, the property that similar nodes are more likely to be friends, elucidate the process of community formation.

Motivated by an immune-response network from biology, we will formalize and explore a new class of networks characterized by a pronounced ring structure to gain insights into systems that are circularly regulated, e.g., when a perturbed ‘healthy’ system needs to move forward to return to the null-state, rather than trying to reverse the damage inflicted. We will study the ring-structure of an unweighted network using two key ideas:

1. Use random walks / diffusion processes to turn a network into a finite metric space.

2. Use the Vietoris-Rips construction to extract persistent homologies from this finite metric space.

After establishing the framework, we will evaluate our methodology on some toy examples and demonstrate that relational data derived from geometric objects can intrinsically store homological information.

**J. Lazovskis: ** This talk will be about how persistence interacts with Morse functions on surfaces, focusing on moduli spaces of Morse–Smale functions on S^2 and their embeddings via different invariants. First we show that all such functions are related by three elementary moves, based on (reverse) cancellation of pairs of critical values. At a finer level, we consider the different ways a Morse–Smale function factors through embeddings into R^3, and uncover the nesting poset of level sets invariant. For persistence as the barcode of the height function, the nesting poset gives insight into the inverse problem of persistence. This is joint work with Mike Catanzaro, Justin Curry, Brittany Fasy, Greg Malen, Hans Riess, Bei Wang, Matt Zabka.

**K. Turner:** The persistent homology transform (PHT) and the Euler characteristic transform (ECT) are two topological transforms that can be viewed as topological analogues of the Radon transform. They provide a way of summarising shapes in a topological, yet quantitative, way. They take a shape, viewed as a tame subset $M$ of $\R^d$, and associates to each direction $v\in S^{d-1}$ a shape summary obtained by scanning $M$ in the direction $v$. These shape summaries are either persistence diagrams or piecewise constant integer valued functions called Euler curves. This talk will explore some recent injectivity results relating to these transforms. An inversion theorem of Schapira implies they are injective on the space of shapes—each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we can also prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, we can construct a finite bound of the number of directions required to specify any shape in certain uncountable families of shapes. The work in this talk is in collaboration with Justin Curry and Sayan Mukherjee.

**T. Needham:** I will introduce the concept of a decorated merge tree (DMT), an invariant which tracks interactions between homological features in multiple degrees for a filtered space. Intuitively, a DMT is a merge tree overlaid with higher dimensional barcodes. Formally, a DMT can be understood abstractly in terms of category theory or concretely as a barcode-attributed combinatorial graph. There is a natural extension of interleaving distance to the setting of DMTs; I will discuss stability properties of this metric as well as methods for computing it via Gromov-Wasserstein distance, a tool from optimal transport. This is joint work with Justin Curry, Haibin Hang, Washington Mio and Osman Okutan.

**A. Gosztolai: ** Defining the geometry of networks is typically associated with embedding in low-dimensional spaces such as manifolds. This approach has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, the choice of embedding space is network-specific, and incompatible spaces can result in information loss. Instead of using embeddings to “geometrise” networks, in this talk, I will consider the Ollivier-Ricci edge curvature, a well-known discrete curvature notion, and extend it using dynamical processes to define the scale-dependent network geometry. I will relate the edge curvature to information spreading through the curvature gap exhibited by the evolution of the curvatures at characteristic timescales that indicate network bottlenecks. Curvature gaps also give a rigorous geometric viewpoint on community detection through the notion of geometric modularity. I will show that optimising the geometric modularity for a graph with a given curvature distribution robustly recovers communities until the phase transition of detectability. I will also consider real-world networks, including the European power grid and the C. elegans homeobox gene regulatory network, finding previously unidentified communities on multiple scales.

**E. Munch: **Merge trees are a type of graph-based topological summary that track the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this talk, we consider the problem of comparing two merge trees via the notion of interleaving distance in the metric space setting. We show that the interleaving distance is intrinsic on the space of labeled merge trees and provide an algorithm to construct metric 1-centers (a kind of averaging procedure) for collections of labeled merge trees. We further prove that the intrinsic property of the interleaving distance also holds for the space of unlabeled merge trees. Our results are a first step toward performing statistics on graph-based topological summaries, and show how these tools can be used for visualization of tree data. This work is joint with E. Gasparovic, L. Yan, S. Oudot, K. Turner, B. Wang, and Y. Wang.

**S. Ebli: ** In this talk I will present a notion directed collapsibility of directed Euclidean cubical complexes. One application of this is in the non-trivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. In our case, the direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex. This is joint work with R. Belton, R. Brooks, L. Fajstrup, B. Fasy, N. Sanderson, and E. Vidaurre.

**A. Elchesen: ** It is well-known that persistence diagrams can be obtained from the rank function via Möbius inversion. In the case of the graded rank function introduced by Betthauser, Bubenik, and Edwards, Möbius inversion gives rise to sets of ordered pairs which may have negative multiplicities. We call these virtual persistence diagrams. We show that the 1-Wasserstein distance extends to virtual persistence diagrams in a natural way. We also consider the measure-theoretic analog of signed measures and distances between them. Motivated by the work of Divol and Lacombe, we set up a framework for discussing the (signed) Wasserstein distances for persistence diagrams and measures defined on arbitrary pairs of metric spaces. Lastly, we will discuss connections to universality of the Wasserstein distances.This is joint work with Peter Bubenik.

**E. Hermansen: ** Grid cells are found in the medial entorhinal cortex of the mammalian brain and are characterized by having spatial firing regions tessellating the environment the animal visits in a hexagonal grid-like pattern. The cells are confined to separate populations called modules, determined by their spatial scaling and orientation. With the recently developed Neuropixel probes, we record the activity of 65-165 cells in six modules across three rats during both free foraging and sleep, and, using persistent cohomology and circular coordinatization as our main tools, we depict toroidal structure of the population activity and decode the time-dependent toroidal position the modules encode. We show that individual cells are preferentially active at singular positions on the torus, and that these positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells.

**G. Grindstaff: **A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose internal edges have a positive weight. Given a set of n leaves, the moduli space of all phylogenetic trees with this leaf set can be endowed with a piecewise-Euclidean metric to form Billera-Holmes-Vogtmann phylogenetic tree space, which is used to define tree statistics. In 2017, Abreu and Pacini showed that the group of cone complex isomorphisms of the moduli space of trees is $S_n$; using combinatorics and CAT(0) and Euclidean geometry, I’ll give an alternate proof and show that all BHV isometries come from the same relabeling maps, even when we don’t require that the cone structure be preserved. This result is relevant to distance-based analyses of sets of phylogenetic trees, and produces a complete description of small neighborhoods of points in tree space.

**C. Xu: **Topological data analysis is an arising new field studying the geometry and topology of data using mainly algebraic topology approaches. While there are already different effective ways to study persistent homology, we take an algebro-geometric perspective to approach it in this research via observing the high similarity between partial flags and persistence modules. We prove a bijective correspondence between persistence diagrams and Schubert cells of an associated partial flag variety. This correspondence provides a passage for transferring the partial order structure and the multiplicative structure from partial flag varieties to persistence modules. It also opens a door for exploring further possibilities of adding more structures on persistence modules.

**I. Voli ć: **Simplicial complexes are versatile objects whose usefulness in real-world applications has increased dramatically in recent years. In this talk, I will discuss a new application where certain types of political systems and their interactions are modeled by simplicial complexes. In this model, agents in a political system are represented by vertices and their interactions by simplices. Some basic topological constructions on simplicial complexes, like the wedge, cone, and homology, then turn out to have interpretations in the world of political structures. I will also introduce the notions of a viability of an agent and the stability of a political structure and see how they interact with the topology of simplicial complexes. Many future potential avenues of investigation will also be pointed out.

**M. Carrière: ** Persistent homology is a common tool of topological data analysis, which aims at computing and encoding the geometry and topology of given datasets. In this talk, I will present a novel application of persistent homology to characterize the spatial arrangement of immune and tumor cells in the context of breast cancer. More specifically, quantitative and robust characterizations are built by computing (multiparameter) persistent homology out of a staining technique (called quantitative multiplex immunofluorescence) which allows to obtain spatial coordinates and stain intensities on individual cells. The resulting persistence modules are then converted into descriptors (persistence diagrams for scalar filtrations, multiparameter persistence images for multiparameter filtrations) and evaluated as characteristic biomarkers of cancer subtype and overall survival. This provides new insights and possibilities on the general problem of building (topology-based) biomarkers for immune responses.

**J. Hansen: **Cellular sheaves are a discrete and computable representation of constructible sheaves over cell complexes, and naturally model many situations where data of varying types is parameterized by a space. Assigning inner products to the stalks of a cellular sheaf allows us to apply the constructions of discrete Hodge theory to a canonically associated cochain complex. We obtain generalizations of the discrete Hodge Laplacians, including the ubiquitous graph Laplacian. These operators have interesting algebraic and spectral properties, and allow us to use the local-to-global structure of cellular sheaves in real-world scenarios where robustness is essential. This talk will outline the construction of cellular sheaves and their Laplacians, discuss the generalization of spectral graph theory to spectral sheaf theory, and sketch avenues for the application of sheaf Laplacians to engineering and scientific problems.