# The Applied Topology Seminar

All talks will be held on Mondays from 3:00 PM until 4:00 PM in Room 4C2 of the David Rittenhouse Lab Building. The organizers are Robert Ghrist and Brendan Fong. To schedule a talk, email brendan.fong at cs.ox.ac.uk. Here is a list of this year's seminar talks from the official Penn math calendar.

# Spring 2016

## 25 Apr 2016: Pablo Camara, Columbia

Speaker: Pablo Camara, Columbia University
Title: Topological Methods for Molecular Phylogenetics

Abstract: The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relations within and across species. Recombination, re-assortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Recently, topological data analysis methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. In this talk I will discuss recent developments in the study of recombination using persistent homology, and I will present several biological applications, including the construction of high-resolution whole-genome human recombination maps.

## 18 Apr 2016: Rachel Levanger, Rutgers

Speaker: Rachel Levanger, Rutgers University
Title: A Comparison Framework for Interleaved Persistence Modules

Abstract: While it is often desired to compute the persistence diagram of a filtration of a topological space precisely, this is routinely not possible for many reasons. First, it might be impossible to encode the exact filtration into a computer. Second, even if this step is accomplished, it might be computationally infeasible to compute the associated persistence diagram (e.g. nerve lemma and Čech complex). As a result, it is commonplace to substitute an approximation (e.g. Vietoris-Rips complex) and use its persistence diagram instead. But what, exactly, have we lost? Most computational topologists are aware of the bottleneck distance between two persistence diagrams, and the literature up until this point typically stops here in terms of error bounds of approximations (or perhaps goes one step further to log-bottleneck). In this talk, we propose a more rigorous framework for analyzing the approximations of persistence diagrams.

## 11 Apr 2016: Omer Bobrowski, Duke

Speaker: Omer Bobrowski, Duke University
Title: Topological Consistency via Kernel Estimation

Abstract: The level sets of probability density functions are of a considerable interest in many areas of statistics, and topological data analysis (TDA) in particular. In this talk we focus on the problem of recovering the homology of level sets from a finite sample. The main difficulty stems from the fact that even small perturbations to the estimated density function can generate a very large error in homology. In this talk we present an estimator that overcomes this difficulty and recovers the homology accurately (with a high probability). We discuss two possible applications of the proposed estimator. The first one is recovering the homology of a compact manifold from a noisy point cloud. The second application is recovering the persistent homology of the super level sets filtration. Finally, we show that similar methods can be used in the analysis of nonparametric regression models.

## 28 Mar 2016: Jeffrey Seely, Columbia

Speaker: Jeffrey Seely, Columbia University
Title: Neural Computation: Visual Cortex versus Motor Cortex

Abstract: Visual cortex and motor cortex lie at opposing ends of the brain’s processing pathway. The goal of visual cortex is to compute representations of its input. The goal of primary motor cortex is to generate movement—the brain’s output. Here, we analyze datasets from both areas. All data is arranged in a neuron by stimulus by time tensor. We show how the ‘tensor structure’ of the data can reveal fundamentally different computational strategies employed by both areas—namely, whether an area encodes external variables or acts as a dynamical system. I will also include my current attempts on applying persistent homology and mapper to the data, as well try to gear the talk toward the question, "How might TDA be helpful for this type of data?"

## 14 Mar 2016: Robert Short, Lehigh

Speaker: Robert Short, Lehigh University
Title: Where Motion Planning and Cohomology Collide

Abstract: Topological robotics is a subfield of applied algebraic topology that concerns itself with motion planning algorithms. It seeks to pin down a value known as the topological complexity for any given space. Loosely, this corresponds to the least number of rules needed to perform continuous motion planning on the space.

While the name topological complexity'' is recent, the ideas making the loose statement precise have been around for decades. Using techniques driven by work from the 1960s, along with more recent principles, it is possible to calculate the topological complexity exactly for many spaces.

In this talk, we will focus primarily on the work of Michael Farber in his book Invitation to Topological Robotics as well as the recent work of Don Davis studying the topological complexity of polygon spaces. In the process, we will examine some of the tools for finding upper and lower bounds for topological complexity, and see the application of them in some nice examples.

# Fall 2015

## 23 Nov 2015: Michael Lesnick, Princeton

Speaker: Michael Lesnick, Princeton University
Title: Interactive Visualization of 2-D Persistent Homology.

Abstract: In topological data analysis, we often study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, a single filtered space is not a rich enough invariant to encode the interesting structure of our data. This motivates the study of multidimensional persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations. The homological invariants of these "multifiltered spaces," while much richer than their 1-D counterparts, are also far more complicated. As such, adapting the usual 1-D persistent homology methodology for data analysis to the multi-D setting requires some some new ideas.

In this talk, I'll introduce multi-D persistent homology and discuss joint work with Matthew Wright on the development of a tool for the interactive visualization of 2-D persistent homology.

## 16 Nov 2015: Safia Chettih, Oregon

Speaker: Safia Chettih, University of Oregon
Title: Topology of Configurations on Trees

Abstract: The homology and cohomology groups of configurations of n unordered points are known on a number of simple graphs, but elude a general combinatorial description. By considering instead a discretized model for configuration spaces of graphs, we can apply a discretized version of Morse theory that simplifies calculations of homotopy type. I will describe how recent results may be extended to ordered configurations and give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on trees, and talk about the geometric and combinatorial structures interrelating configurations on graphs.

## 09 Nov 2015: John Meier, Lafayette

Speaker: John Meier, Lafayette College
Title: Topology at Infinity for Cube Complexes

Non-positively curved cube complexes arise in an impressive number of applied and theoretical settings, including configurations on graphs and right-angled Artin groups. The topology at infinity for an unbounded, locally finite complex is the topology that persists in the complement of any finite sub-complex. For example, a graph is one-ended if there is a single connected, unbounded component in the complement of any finite sub-graph. I will survey joint work with Jon McCammond and Noel Brady on local-to-asymptotic results for non-positively curved cube complexes, and explain in some detail joint work with Liang Zhang in the case of configurations on a complete graph.

## 02 Nov 2015: Randall Kamien, Penn

Speaker: Randall Kamien, University of Pennsylvania
Title: Linked or Not So Much?

Abstract: The order parameter of the smectic liquid crystal phase is the same as that of a superfluid or superconductor, namely a complex phase field. We show that the essential difference in boundary conditions between these systems leads to a markedly different topological structure of the defects. Screw and edge defects can be distinguished topologically. This implies an invariant on an edge dislocation loop so that smectic defects can be topologically linked not unlike defects in ordered systems with non-Abelian fundamental groups.

## 26 Oct 2015: Rory Conboye, Florida Atlantic

Speaker: Rory Conboye, Florida Atlantic University
Title: Toward a Piecewise-Linear Analogue of Ricci Curvature

In three dimensions, a simplical piecewise-linear manifold is formed by joining flat Euclidean tetrahedra. The triangular faces that border two neighbouring tetrahedra are identified, so that the space formed by any pair of tetrahedra is also Euclidean. Curvature then arises when the developement of tetrahedra is continued around an edge, with the sum of the dihedral angles at the edge deviating from a full rotation by a 'deficit angle'. These deficit angles are related to a projection of the Riemann curvature tensor, making local definitions of the Riemann and Ricci curvature tensors quite difficult. This talk will cover some recent work on relating the deficit angles to sectional curvature, and combining this with a piecewise-linear scalar curvature to give the a Ricci curvature in the direction of each edge. Potential extensions to higher dimensions will also be discussed.

## 19 Oct 2015: Subhrajit Bhattacharya, Penn

Speaker: Subhrajit Bhattacharya, University of Pennsylvania
Title:: Path Homotopy Invariants and their Application to Optimal Trajectory Planning

I will consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in applications to robotics, homotopy path-planning in applications usually focuses on subsets of the Euclidean plane. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this talk I will propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. I will also present explicit results that apply to knot and link complements in 3-space, and discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension.

## 05 Oct 2015: Sara Kališnik Verovšek, Stanford

Speaker: Sara Kališnik Verovšek, Stanford University
Title: Parametrized Homology and Parametrized Alexander Duality

Abstract: An important problem with sensor networks is that they do not provide information about the regions that are not covered by their sensors. If the sensors in a network are static, then the Alexander Duality Theorem from classic algebraic topology is sufficient to determine the coverage of a network. However, in many networks the nodes change position over time. In the case of dynamic sensor networks, we consider the covered and uncovered regions as parametrized spaces with respect to time. I will discuss parametrized homology, a variant of zigzag persistent homology, which measures how the homology of the level sets of a space changes as the parameter varies. I will show also how we can extend the Alexander Duality theorem to the setting of parametrized homology.

# Spring 2015

## 27 Apr 2015: Michael Robinson, American University

Speaker: Michael Robinson, American University
Title: Quasiperiodicity: Processing Signals using Topology

Abstract: When can you infer the state of a system from measurements of a signal? In a surprisingly diverse set of situations, rather precise bounds can be obtained on the number of measurements needed to constrain a system from the Whitney embedding theorem. This result of differential topology is easy to state, easy to use, and intuitively satisfying. It is also remarkably easy to demonstrate with simple consumer-grade equipment, as I'll explain.

When the state space is small, then the system has a canonical representation. The general theory and canonical representation leads to robust, practical nonlinear "topological filters", which generalize the linear filters already used extensively. Because they are built topologically, the local structure of these filters can be tailored easily and provides a solid theoretical grounding for nonlinear matched filters. I will advocate for the wider application of topological methods within engineering, and show how topological filters can improve the quality of maritime radar images.

## 20 Apr 2015: Patrick Hafkenscheid, Vrije Universiteit

Speaker: Patrick Hafkenscheid, Vrije Universiteit Amsterdam
Title: Braid Morse Homology and its Computations

Abstract: Braid Morse Homology aims to be a bridge between the study of certain parabolic PDEs and Algebraic Topology. The idea is to extend finite dimensional Morse Theory to the case of certain objects known as braids. Braids are in some sense a generalisation of periodic functions that allows us to consider multiple periodic functions simultaneously. The Homology theory can give some nice forcing results on periodic solutions of the PDE.

The problem with the Braid Morse Homology is that it is in general very hard to compute, however a very nice discretisation makes it so we can actually compute these objects! In this talk I will discuss braids, a very short introduction to Morse Homology and explain in what way we can compute the object.

## 23 Mar 2015: Iris Yoon, Penn

Speaker: Iris Yoon, University of Pennsylvania
Title: Sperner’s Lemma via Cellular Sheaf Cohomology

Abstract: Sperner’s lemma is a combinatorial result that is equivalent to the Brouwer fixed point theorem. We can consider a Sperner coloring as data parametrized by a simplex, and hence use sheaf theory to collate the data. We will first discuss constructs of cellular sheaves and cellular sheaf cohomology. Results from cellular cohomology generalize to analogous results in cellular sheaf cohomology. We will prove Sperner’s lemma using cellular sheaf cohomology and close with a discussion of other lemmas and theorems that we can revisit with the perspective of sheaf cohomology, such as Brouwer fixed point theorem and Helly’s theorem.

## 23 Feb 2015: Greg Henselman, Penn

Speaker: Greg Henselman, University of Pennsylvania
Title: Cellular Matroids and Topological Data Analysis

Abstract: A matroid on finite ground set $$E$$ is a combinatorial abstraction of a function from $$E$$ to a finite-dimensional vector space. The first portion of this talk will cover basic constructs and results in the theory of matroids, and highlight some old (but powerful) applications in topology, optimization, and combinatorics. Of particular interest are the matroids represented by boundary operators of CW complexes; as we will see, invariants of these combinatorial objects encode useful information about the underlying spaces, and the specialization of several matroid algorithms to this context offers efficient partial solutions to problems, such as deciding embeddability, for which no viable alternatives are known*. We will close with a brief discussion on combinatorial $$k$$-connectivity, topological clustering, and proposed applications in robotics & neuroscience.

* To us.

## 16 Feb 2015: Michael Erdmann, Carnegie Mellon

Speaker: Michael Erdmann, Carnegie Mellon University
Title: Geometry and Topology of Privacy

Abstract: An old problem in privacy research is understanding how publication of an anonymized relation can lead to privacy loss via background or auxiliary information. This talk examines that problem from the perspective of Dowker's duality theorem for relations. The relation itself defines two simplicial complexes: a space of attributes and a space of individuals. Dowker's theorem says that these two spaces are homotopy equivalent. That equivalence suggests two definitions of privacy: attribute-privacy and association-privacy. The talk discusses that construction and surveys some consequent results.

## 26 Jan 2015: Brendan Fong, Oxford

Speaker Brendan Fong, University of Oxford
Title: The Black Box Functor: Compositional Analysis of Passive Linear Networks

Abstract: There is an obvious notion of composition for electrical circuits: just solder some terminals together. But what how do we compute the behaviour of the composite system from an understanding of its parts, and exactly what algebraic structure is present? Viewing a passive linear circuit as an object that constrains the current-potential readings at its terminals to lie in a specific linear subspace, in this talk we will discuss how to construct a monoidal functor from a category of circuit diagrams to the category of linear relations that expresses this semantics. This represents the beginning of a categorical account of Willems' behavioural approach to open and interconnected systems.

This is joint work with John Baez.

# Fall 2014

## 01 Dec 2014: Peter Bubenik, Cleveland State

Speaker: Peter Bubenik, Cleveland State University
Title:: Categorical Topological Data Analysis

Abstract: In this talk, I will show how the language of category theory can be used to shed light on one of the main tools in applied topology, persistent homology.

## 24 Nov 2014: Carlos Cadavid, EAFIT

Title : Minimal Morse Functions via the Heat Equation in Homogeneous Riemannian Manifolds

Abstract: Let $$(M,g)$$ be a closed Riemannian manifold that is homogeneous, in the sense that each pair of points have mutually isometric neighborhoods, and let $$\Delta_ g$$ be its Laplace-Beltrami operator. The heat equation on $$(M,g)$$ is $\frac{df}{dt}=-\Delta_g(f)$ and for each initial condition $$u$$ in $$L^2(M)$$ there exists a unique solution $$f_t(.):=f(.,t)$$ satisfying $$f_0=u$$. It has been observed in several examples that for generic $$u$$ and large enough $$t$$, $$f_t$$ is a Morse function having the least number of critical points admitted by any Morse function on $$M$$. In this talk I will quickly review the definitions and facts necessary to state the previous observation as a formal conjecture, and then I will mention the examples that have been tested so far.

## 03 Nov 2014: Henry Adams, Duke

Title: The Vietoris-Rips Complex of the Circle

Abstract: Given a metric space and a positive connectivity parameter, the Vietoris-Rips simplicial complex has a vertex for each point in the metric space, and contains a set of vertices as a simplex if its diameter is less than the connectivity parameter. A theorem of Jean-Claude Hausmann states that if the metric space is a Riemannian manifold and the connectivity parameter is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. What happens for larger connectivity parameters? We show that as the connectivity parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy type of the circle, the $$3$$-sphere, the $$5$$-sphere, the $$7$$-sphere, ..., until finally it is contractible. Joint work with Michal Adamaszek, Florian Frick, Christopher Peterson, and Corrine Previte.

## 13 Oct 2014: Paul Bendich, Duke

Speaker: Paul Bendich, Duke University
Title: Multi-scale Looping and Branching Analysis of Brain Artery Trees

Abstract: The statistical analysis of a population of brain artery trees is considered. New representations of these tree structured data objects are developed, using ideas from topological data analysis. Specifically, a number of representations of each brain tree, using persistence diagrams that quantify branching and looping at multiple scales, are considered. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to improved correlations with covariates such as age and sex, relative to earlier analyses of this data set. Strikingly, these correlations remain strong even after controlling for more obvious geometric differences in the set of trees.

This is joint work with Alex Pieloch, J.S. Marron, Ezra Miller, and Sean Skwerer, and the dataset was obtained from the lab of Elizabeth Bullitt at UNC-CH.

## 15 Sep 2014: Thanos Gentimis, NCSU

Speaker: Thanos Gentimis, North Carolina State University
Title: Directed Persistence

Abstract: Given a finite point set $$X$$, one constructs the Čech complex at scale $$r$$ by taking the nerve of the subset covered by $$r$$-balls centered at the points of $$X$$. Given a direction vector $$u$$ and a scale $$0 \lt c \lt 1$$, we construct a cover by ellipsoids with major axis parallel to $$u$$ and length $$r$$. The normal directions to $$u$$ have length $$cr$$. We then compute the homology of the complex obtained as the nerve of this cover. This additional structure defines a generalized persistence module, yields a correspondence between rotations and interleavings of persistence modules and suggests an approach to detect directed homological features using maximum persistent length. This is joint work with Dr. Greg Bell from UNC Greensboro.

## 08 Sep 2014: Chad Giusti, Penn

Speaker: Chad Giusti, University of Pennsylvania
Title: Topological Analysis of Symmetric Matrices

Abstract: It is common in biological applications to work with data presented as correlation matrices, and it is often the case that the observed variables are distorted by an unknown monotonic nonlinearity. In such settings, the usual linear algebra-based tools for analysis of matrices can fail to detect structure of interest, or even give false impressions of its existence.

By analogy to the Jordan canonical form, we introduce the "order canonical form" of a symmetric matrix, which retains all of the invariant information in the matrix under an element-wise action of the group of monotonic increasing functions. Using the tools of persistent homology, we extract from the order complex a family of signatures of geometric structure (or lack thereof) in the elements of the matrix. As an application, we study the pairwise correlations in the neuronal population of the hippocampus in rats under a variety of behavioral conditions and find strong evidence of a geometric structure in the functional connectivity of the network.