Applied Topology and Geometry Seminar
Date  Speaker  Title and Abstract 

May 3  Benjamin Kimia  Similarity Manifold of Shapes and Images
Similarity is a key concept in organizing a representation space in computer vision. Two examples are illustrated here, one for organizing the shape space and one for recovering image correspondence. In both examples, it is argued that the topological representation of shapes and images does not require dimensionality reduction per se. Rather, it is important to capture the underlying manifold, especially when millions of exemplars are available. This emphasis requires incremental construction and efficient indexing methods that are suitable for a very large dataset. This talk aims to stimulate a discussion of possible paths forward. 
April 26  Carina Curto  Graphs, neural networks, and emergent dynamics in the brain
Many networks in the brain display internallygenerated patterns of activity  that is, they exhibit emergent dynamics that are shaped by intrinsic properties of the network rather than inherited from an external input. While a common feature of these networks is an abundance of inhibition, the role of network connectivity in pattern generation remains unclear. In this talk I will introduce Combinatorial ThresholdLinear Networks (CTLNs), which are simple "toy models" of recurrent networks consisting of thresholdlinear neurons with effectively inhibitory interactions. The dynamics of CTLNs are controlled solely by the structure of an underlying directed graph. By varying the graph, we observe a rich variety of nonlinear dynamics including: multistability, neuronal sequences, and complex rhythms. These patterns are reminiscent of population activity in cortex, hippocampus, and central pattern generators for locomotion. I will present some new theorems about CTLNs, and explain how they allow us to predict features of the dynamics by examining properties of the underlying graph. 
April 5 *This talk will be at 3pm in Barus and Holley 158 *  Lori Ziegelmeier 
Persistence Images: A Stable Vector Representation of Persistent Homology
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finitedimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vectorbased machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, we present novel applications arising from dynamical systems which reveal the discriminatory power of PIs. 
March 22 *CANCELLED*  Konstantin Mischaikow  Reduction and Reconstruction of Complex SpatioTemporal Data
It is almost cliche at this point to note that high dimensional data is being collected from experiments or generated through numerical simulation at an unprecedented rate and that this rate will continue rising extremely rapidly for the foreseeable future. Our interest is in data associated with high dimensional nonlinear complex spatiotemporal dynamics. The focus of this talk is on our efforts to use persistent homology both as a dimension reduction technique and a technique for reconstructing structures of the underlying dynamical system. I will present some results associated with dynamics of fluid convection and dense granular media and will try to highlight open questions. 
March 15  Andrew Blumberg  Approximating the GromovHausdorff distance
Distances between point clouds play an important role in the topological and geometric analysis of data. A theoretically important metric is the GromovHausdorff distance; however, this is not usually feasible to compute. In this talk, I will give an overview of standard approaches to finding tractable distances and then describe some recent work with Villar, Bandeira, and Ward that applies techniques from convex optimization. 
Date  Speaker  Title and Abstract 

December 7  Pablo Camara  A topological approach to the identification of
elusive cancer genes
Developing targeted cancer therapies requires the stratification of cancer patients according to the molecular alterations that are relevant for tumor progression. Commonly used methods for identifying cancerassociated mutations seek signatures of positive selection based on recurrence across large cohorts of patients. However, these strategies have limited power to identify functional variants that occur at low frequencies or in hypermutated tumors. Building upon Mapper representations, we have devised a general unsupervised framework for feature selection. In this talk, I introduce the general framework and present its application to the identification of cancerassociated mutated genes. Our goal is to identify new potential therapeutic targets that, because of their low prevalence, have remained elusive to current methods of detection. 
November 2  Anthea Monod  Tropical Sufficient Statistics for Persistent Homology
with a Parametric Application to Infectious Viral Disease
In this paper, we show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes  multiscale summaries of topological characteristics that capture the "shape" of data, but have complex structures and are therefore difficult to use in statistical settings. Our sufficiency result allows for the assumption of classical probability distributions on Euclidean representations of barcodes. This in turn makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations. In particular, we show that exponential family distributions may be assumed, and that likelihoods for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to HIV and avian influenza data. 
October 26 ** This talk will be at 4pm in the Foxboro Auditorium ** 
Attila Gyulassy  Feature Driven Exploration of Images Using Topology
Two and threedimensional images arise in a multitude of disciplines, from computer simulations to experimental sources such as microscopy. A key step to scientific discovery is extracting semantic information from image data. This talk presents practical application of Morse theory to extract and reason with features intrinsic to images. We will discuss both practical algorithms for computing the MorseSmale complex as well as realworld applications drawing on imaged and simulated examples from material science, basic energy research, geologic exploration, and neuroscience. 
October 19  Lorin Crawford  Functional Data Analysis using a Topological Summary
Statistic: the Smooth Euler Characteristic Transform
In this talk, we introduce a novel statistic, the smooth Euler characteristic transform (SECT), which is designed to integrate shape information into regression models by representing shapes and surfaces as a collection of curves. Due to its welldefined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. We illustrate the utility of the SECT in a radiomics context by showing that the topological quantification of tumors, assayed by magnetic resonance imaging (MRI), are better predictors of clinical outcomes in patients with glioblastoma multiforme (GBM). We will close by showing that SECT features alone explain more of the variance in patient survival than gene expression, volumetric features, and morphometric features. 
October 5  Elchanan Solomon  A HomologyBased Metric on the Space of Metric Graphs
In order to find structure in a large data set it is important to be able to compare data points. For example, if Netflix wants to recommend a movie to you that is popular with similar users, it must first decide what it means for two users to be similar. For data sets consisting of vectors or matrices there are many good ways of making such comparisons. However, a lot of data comes to us in the form of a shape: e.g. a social network graph, an MRI scan, a triangulated 3D model. How can we reasonably and effective compare these shapes? The goal of this talk is to introduce an appliedtopologybased approach for comparing graphs that was first considered in a paper of Dey, Shi, and Wang in 2015. This will require a quick introduction to metric geometry and persistent homology for which we will assume no background. Their comparison metric enjoys good theoretical and computational properties, however it is possible for it to conclude that two distinct graphs are distance zero from each other. We conclude the talk with a recent result of OudotS. (paper in preparation) that this generically does not happen. 
September 21  Justin Curry  The Fiber of the Persistence Map
The persistence map is the map that sends a function on a topological space to it's collection of persistence diagrams, which are canonical invariants of filtering a space by sublevel sets and taking homology in each degree. Geometrically, a persistence diagram is simply a configuration of points in the plane. In this talk I will study which configurations of points are possible and what the ramification of this map is for the simplest possible casefunctions on the  interval. This talk is based on a recent arXiv preprint (arxiv.org/abs/1706.06059) and will require minimal background to understand. 
Date  Speaker  Title and Abstract 

April 27  Ellen Gasparovic  A PersistenceBased Summary for Metric Graphs
In this talk, we will focus on giving a qualitative description of information that one can capture from metric graphs using a certain topological summary. In particular, we will give a complete characterization of the 1dimensional persistence diagrams for metric graphs under a particular intrinsic setting. Timepermitting, we will also look at two persistencebased distances that one may define for metric graphs and discuss progress toward establishing their discriminative capacities. 
April 20  Sayan Mukherjee 
The Geometry of Synchronization Problems and Learning Group Actions
We develop a geometric framework that characterizes the synchronization problem  the problem of consistently registering or aligning a collection of objects. The theory we formulate characterizes the cohomological nature of synchronization based on the classical theory of fibre bundles. We first establish the correspondence between synchronization problems in a topological group G over a connected graph Γ and the moduli space of flat principal Gbundles over Γ, and develop a discrete analogy of the renowned theorem of classifying flat principal bundles with fixed base and structural group using the representation variety. In particular, we show that prescribing an edge potential on a graph is equivalent to specifying an equivalence class of flat principal bundles, of which the triviality of holonomy dictates the synchronizability of the edge potential. We then develop a twisted cohomology theory for associated vector bundles of the flat principal bundle arising from an edge potential, which is a discrete version of the twisted cohomology in differential geometry. This theory realizes the obstruction to synchronizability as a cohomology group of the twisted de Rham cochain complex. We then build a discrete twisted Hodge theory  a fibre bundle analog of the discrete Hodge theory on graphs  which geometrically realizes the graph connection Laplacian as a Hodge Laplacian of degree zero. Motivated by our geometric framework, we study the problem of learning group actions  partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations. A dual interpretation is to learn finitely generated subgroups of an ambient transformation group from noisy observed group elements. A synchronizationbased algorithm is also provided, and we demonstrate its efficacy using simulations and real data. joint work with: Tingran Gao and Jacek Brodzki 
April 13  Zvi Rosen 
From Combinatorial Codes to Realizing Arrangements
A combinatorial neural code is a set of 01 vectors (codewords) derived from the nonempty noncovered regions of a collection of sets in R^n. Inspired by applications, we study arrangements whose sets are (a) convex, (b) halfspaces, and (c) intervals. Our question: How do topological properties of the arrangement translate into combinatorial properties of the code? We use tools from commutative algebra, topology, and discrete geometry to place some necessary conditions and describe the minimal dimension of a realizing arrangement. Based on joint work with C Curto, E Gross, V Itzkov, J Jeffries, A Kunin, K Morrison, M Omar, A Shiu, N Youngs, and Y Zhang. 
April 13  Carl McTague  A New Approach to Euler Calculus
I will describe a new approach to Euler calculus—a subject rooted in algebraic geometry which has recently found surprising applications to data analysis. The new approach relies on differential geometry and extends Euler calculus to continuous integrands. It requires a metric but is nevertheless defined within any Ominimal theory. It satisfies a Fubini theorem and extends to a functor. Passing to the “adiabatic limit” recovers the classical Euler calculus. This suggests new applications of differential geometry to data analysis. 
April 6  Yitzchak Solomon 
Spectral Methods in Computational Geometry
Spectral geometry aims to study the geometry of a Riemannian manifold through the eigenvalues and eigenfunctions of various differential operators. The case of the LaplaceBeltrami operator has been the most wellstudied, and is related to the heat flow on the manifold. Although one cannot always "hear the shape of the drum", as Weyl conjectured, this spectrum contains a lot of geometric information. Recent work of Memoli and others have shown how to use this spectral data to define a metric on the space of shapes, as well as effectively construct isometries and approximate isometries. In this talk we will review the basics of spectral geometry and survey some of these referenced applications. 
March 20 (Monday!!)  Greg Henselman 
Combinatorial Homology: A Simplified Approach to Persistence and Computation, via Matroids
A basic notion in modern topological data analysis (TDA) is the idea of a persistent homology group. The elements of homological persistence have existed since at least the mid 1980’s, emerging independently in the work of Robins in shape theory, of Edelsbrunner, Kirkpatrick, and Seidel in biogeometry, and of Frosini and Ferri in size theory. Two principle challenges in modern TDA center on questions of modeling and computation: what do persistent homology groups have to say about empirical data, and how can they be computed? A recurring issue in the face of these challenges has been the body of algebraic machinery needed to make the idea of persistent homology precise; this apparatus, while standard in pure mathematics, poses a basic barrier to entry for newcomers to the field of TDA, and an impediment to formal interpretation of elementary topological statistics. In this talk we introduce a new approach to persistent homology which relies on no algebraic structure whatever, but rather some elementary combinatorial properties of linear independence. This simplified framework yields useful insights into the mathematical descriptors commonly associated to homological persistence  barcodes and generators  as well as computation. As an application, we will discuss a new algorithm to compute barcodes and generators, recently implemented in the Eirene library for computational persistence, and its advantages for large and highdimensional data sets. 
March 16  Henry Adams  The theory of VietorisRips complexes
Given a metric space X and a distance threshold r > 0, the VietorisRips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of JeanClaude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the VietorisRips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of VietorisRips complexes for larger values of r, even though these complexes arise naturally in applications of using persistent homology. We describe how as r increases, the VietorisRips complex of the circle obtains the homotopy types of the circle, the 3sphere, the 5sphere, the 7sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, certain VietorisRips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite VietorisRips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. With this new metric, we describe the first change in homotopy type (as r increases) of VietorisRips complexes of higherdimensional spheres. Joint work with Michal Adamaszek, Florian Frick, and Samadwara Reddy. 
March 9  Liz Munch 
The Convergence of Mapper
Mapper, a powerful tool for topological data analysis, gives a summary of the structure of data with respect to a filter function and a cover of the function range. Assuming that this data comes from a true, underlying (but possibly not accessible) topological space, a related construction, the Reeb graph, can be thought of as the ground truth and Mapper, its approximation. In particular, working with a better data sample and/or a more refined cover intuitively results in a Mapper graph which is more similar to the Reeb graph. In this talk, we will discuss a method for defining a distance on these objects via the interleaving distance idea from persistent homology. We can look at various ways to rigorously quantify the idea that Mapper converges to the Reeb graph, as well as ideas for approximation methods of the distance itself. 
March 2  Elina Robeva 
Orthogonal Tensor Decomposition
A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tightframe decomposable tensors and study their properties. 
February 23  Don Sheehy  What Can Persistent Homology Tell Us About Frechet Distance?
We show how to relate the Frechet distance between curves to the bottleneck distance between the persistence barcodes of distance functions induced by the curves. Specifically, we show how computing some persistencebased signatures can yield lower bounds on the Frechet distance. This approach is very general and applies to surfaces and other images of topological spaces into metric spaces. 
February 23  Rachel Levanger  Studying fluid dynamics with persistent homology
Problems in fluid dynamics are notoriously challenging. While models exhibiting a flow trajectory approaching some fixed state are now relatively well understood, systems that are driven far from equilibrium thwart the efforts of many classical analytical methods. Furthermore, simulations are often performed using periodic boundary conditions, introducing symmetries in the system that must be quotiented out in order to study the underlying dynamics, a formidable task when dealing with data lying in such a highdimensional space (the space of ideal solutions of a fluid flow is infinitedimensional, and even a 400x400 pixel image approximation lives in a 160,000 dimensional space!). In this talk, we will show how persistent homology addresses many of these problems simultaneously, using it as a tool for dimension reduction, shape detection, data exploration, and pattern recognition in a suite of examples: Kolmogorov and RayleighBenard convection flows on a twodimensional domain, and combustion simulations on a threedimensional domain. 
February 16  Ann Sizemore  Cliques and Cavities in the Human Connectome
Encoding the axon bundles between brain regions as a complex network has provided novel insights into brain function and disease. Standard network tools describe local or aggregate global phenomena, however many neural functions occur at the mesoscale, involving complex patterns of interactions between multiple brain regions. Here we employ persistent homology to illuminate essential cycles within the structural brain network which may allow for distributed computations. We present longlived cycles in multiple dimensions, and demonstrate the nonrandom nature of these persistent features across eight healthy individuals. Finally we propose implications of these persistent cycles for cognitive function and highlight the potential of this topological view of the brain network. 
February 9  Brown University is closed because of a snowstorm.
 
February 2  David Damiano  A Topological Analysis of SPECT Images of Murine Tumors
In this talk we employ computational topology methods to quantify heterogeneous uptake behavior across time series of singlephoton emission computed tomography (SPECT) images of murine tumors. This behavior cannot be captured by standard aggregate measures such as per cent injected dose per gram or tumortoheart ratio. In particular, we utilize zeroth order persistence diagrams as well as the novel concept of childhood diagrams. Statistical methods are applied to time series persistence and childhood diagrams to detect heterogeneity of uptake within and across study groups in two studies. This behavior is explained in terms of the underlying biological mechanisms. 
Date  Speaker  Title and Abstract 

December, January  No seminar.  Happy Holidays!

December 1  Michael Lesnick  An Introduction to Multidimensional Persistent Homology
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 1D counterparts, are also far more complicated. As such, adapting the usual 1D persistent homology methodology for data analysis to the multiD setting requires some some new ideas. In this talk, I'll introduce multiD persistent homology and discuss some recent progress on this topic. 
November 24  No seminar this week  Happy Thanksgiving!

November 17  Vladimir Itskov  Convex codes, Dowker complex, and detecting nonlinear matrix factorizations.
Many neural systems generate neural activity that can be characterized mathematically as convex codes. A convex code is a subset of the power set $2^{[n]}$ that arises from intersection patterns of convex sets in a Euclidean space. These codes retain topological features of the underlying space of stimuli. Not all codes are convex. What combinatorial properties of a code determine its convexity and what determines its embedding dimension is only partially understood. I will begin by reviewing our recent results on convex codes and the special case of hyperplane codes. I will then explain how hyperplane codes are related to the problem of detecting a matrix factorization, hidden by a nonlinearity. I will also present some computational results and end by describing an approach for detection of convexity and dimension bounds of a noisy neural code. 
November 10  Steve Oudot  A theoretical framework for the analysis of Mapper
Mapper is among the most widely used tools from Topological Data Analysis in the applied sciences and industry. Its output takes the form of a graph whose vertices represent homogeneous subpopulations of the input data. Its main asset for exploratory data analysis lies in its ability to highlight patterns that other methods fail to detect in the data. However, the inherent instability of its output makes it a rather unreliable tool in practice. In this talk I will present some recent contributions to the study of the structural properties of the Mapper graphs. Drawing connections to Reeb graphs and extended persistence theory, I will introduce descriptors for the Mapper graphs, which turn out to be both stable and locally complete. These descriptors can be exploited in the development of a statistical framework to assess the stability of the output of Mapper and to select its input parameters automatically. 
November 3  Yuliy Baryshnikov  Euler Characteristics of Exotic Configuration Spaces
I will describe generating functions for Euler characteristics of multicolored versions of configuration spaces (generalizing a result by S. Gal) and will explain how and where they can be applied. 
October 27  Chad Giusti  What else can persistent homology see?
The usual framework for TDA takes as its starting point that a data set is sampled (noisily) from a manifold embedded in a high dimensional space, and provides a reconstruction of topological features of that manifold. However, the underlying algebraic topology can be applied to data in a much broader sense, carries richer information about the system than just the barcodes, and can be finetuned so it sees only features of the data we want it to see. I will discuss this perspective broadly, and then focus on few particular approaches with applications to neuroscience. 
October 20  Rade Zivaljevic  Algebraic topology and cooperative games
In cooperative games `players' create coalitions in order to achieve some common goal (voting games, profit games, `simple games' of von Neumann and Morgenstern, etc.). We study and compare the simplicial complexes arising in cooperative game theory (threshold complexes, simple games) with the complexes arising as obstructors for embedding (mapping) spaces into higher dimensional euclidean spaces without double (multiple) points (Kuratowski graphs, TverbergVan KampenFlores obstructions, runavoidable complexes (GromovBlagojevicFrickZigler reduction), etc.). 
October 13  Anthea Monod  A Case for Topology and Data
In this talk, I will discuss the relevance of topology in statistics and probability theory. I will give an overview of research topics stemming from applied and random topology, a relatively new field resulting from the application of pointset and algebraic topological methods to data that arise from random processes. The use of topological methods to analyze data is particularly applicable to information technology and networks (e.g. analyzing internet traffic and communications systems), physics (e.g. analyzing cosmological data), biology (e.g. brain imaging and studying gene regulation), and engineering (e.g. studying coordinated robotics). From the perspective of statistics, such methods are applicable to data mining and image processing. I will outline the methodology of topological data analysis, and discuss recent results on the algebraic topology of random fields and complexes, which are important settings in statistics. I will then present some problems that are currently being studied. In particular, I will underline some contributions of applied topology to resolving statistical problems, and how statistical methodology can contribute to resolving applied topological problems. 