Speaker: Henry Adams, Colorado State University
Title: Geometric complexes in applied topology
Abstract: The course will begin with an introduction to applied an computational topology. The shape of a dataset often reflects important patterns within. One such dataset with an interesting shape is the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce persistent homology as a topological tool for learning properties of a space from only a finite sample, and hence for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
Equipped with this motivation, we will move to a mathematical study of geometric complexes arising in applied topology, including nerve, Cech, alpha, and Vietoris-Rips complexes. These complexes thicken a finite sampling into a simplicial complex approximating the underlying space. We will begin by studying these complexes with first principles, but will advertise many open questions and current research directions throughout.
Speaker: Hal Schenck, Iowa State University
Title: Algebraic Tools in Multiparameter Persistent Homology
Abstract: This mini course will consist of 3 talks on Multiparameter Persistent Homology (MPH), with all concepts illustrated with lots of examples, and computational exercises as part of the course.
First Talk: We’ll begin with a brief review of the “classical” case of one-parameter persistent homology, then give an in depth discussion of the tools from algebraic geometry and commutative algebra that play a role in the MPH case: Noetherian rings, associated primes, primary decomposition of modules, localization.
Second Talk: This will continue the theme of the first talk, developing the algebraic tools needed to analyze MPH. We introduce graded and multigraded rings and modules, the Hilbert function, polynomial, and series, and local cohomology in the first half of the talk. The second half of the talk will introduce the multi filtered chain complexes defined by Carlsson-Zomorodian, with a detailed analysis of several examples.
Third Talk: In this talk, we apply the tools from homological algebra and algebraic geometry to analyze MPH. The surprise is that most of the results we obtain are due to the fact that MPH is a fine (Z^n) graded object, which imposes major constraints. If time permits, we’ll see how to use the Grothendieck spectral sequence to probe the structure of MPH, and close by discussing directions for future research.
Speaker: Erika Roldán, The Ohio State University
Title: Random Cubical Complexes
Abstracts: In this mini-course, we will study different stochastic models of cubical complexes. For the most part, we will analyze topological and geometric properties of pure, strongly connected, random 2D and 3D cubical complexes with both uniform and percolation distributions. We will compare the results obtained with the topological and geometric properties observed in cell growth stochastic processes (that are also defined using cubical complexes as cells). This mini-course will have a strong component of discrete and computational geometry, low dimensional topology, and combinatorics.
Speaker: Stefania Ebli
Title: A Notion of Harmonic Clustering in Simplicial Complexes
ABSTRACT: Real-world data, such as biological measurements, naturally lie on a graph, whose edges encode the pairwise relations between the measured signals. Traditionally, the network underlying the data has been studied through the graph Laplacian, L, and its spectrum. The eigenvalues and eigenvectors of the graph Laplacian encode valuable information about the intrinsic structure of the network. For instance, the number of connected components of the graph is equal to the multiplicity of the zero-eigenvalue of L. Moreover, the spectrum of the graph Laplacian has been widely used for clustering points in datasets through the spectral clustering algorithm.
The main goal of this talk is to present the harmonic clustering algorithm, a higher dimensional version of the spectral clustering algorithm. We will first explore the graph Laplacian and the spectral clustering algorithm. Then, we will investigate the properties of the eigenvectors of the Laplacian of order k > 0, focusing mainly on the eigenvectors associated to the zero eigenvalues. Finally we will present the harmonic clustering algorithm, a higher dimensional clustering algorithm, which extract topological features from a dataset using the zero-eigenvectors of the Laplacian. We will show different examples of datasets on which the algorithm has been successfully tested. This is joint work with Gard Spreeman.
Speaker: Raffaella Mulas
Title: Spectrum of the Laplace Operator for Random Geometric Graphs
Abstract: We discuss the spectrum of the normalized Laplace operator for random geometric graphs in the thermodynamic regime. This is a joint work with Antonio Lerario.
Speaker: Dejan Govc
Abstract: In topological analysis of neural systems, data often comes in the form of directed graphs. To such a graph, we typically associate a directed flag complex, which is an ordered simplicial complex consisting of the directed cliques (i.e. transitive tournaments) of the graph. I will talk about a related construction, where instead of directed cliques, arbitrary tournaments are considered. The resulting complex is called the flag tournaplex. We will also consider complexes of tournaments (tournaplexes) more generally. A particularly nice property of these is that they come naturally equipped with certain directionality filtrations which contain information about the underlying structure of the graph. The basic ideas will be explained together with some examples and applications. This is joint work with Ran Levi and Jason Smith.
Speaker: Takashi Owada
Title: Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex
Abstract: Previous attempts to extend the Erdos-Renyi graph to higher dimensions have already given rise to some interesting random simplicial complexes. While the topological study of these models is non-trivial and has led to several seminal works, their applicability is limited by the fact that the randomness in most of these models is governed by a single parameter. In this work, we aim to extend these studies first to the recently introduced multi-parameter random simplicial complex and then, and more importantly, to its dynamic analogue which we introduce here for the first time. In this dynamic setup, the temporal evolution of simplicies of various dimensions is determined by a stationary and possibly non-Markovian process with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our primary results include functional strong laws of large numbers and functional central limit theorems for the Euler characteristic and the Betti number in the critical dimension. The presence of such a dominating dimension was first noted in the single parameter random clique complex case. It is, therefore, pleasing to see that the behavior of topological invariants in the dominating dimension extend naturally to the multi-parameter setup as well.
Joint work with Gennady Samorodnitsky (Cornell) and Gugan Thoppe (Duke).
Speaker: Parker Edwards
Title: Graded persistence diagrams and persistence landscapes
Abstract: Persistent homology computations summarize input data as a persistence diagram, which can be subsequently transformed into a vector for use with machine learning methods. I will introduce graded persistence diagrams, and explain their close correspondence with a particular vectorization scheme: persistence landscapes. This correspondence clarifies the theoretical connections between landscapes, persistence diagrams, and rank functions. I will also discuss a suitable generalization of the 1-Wasserstein distance for graded diagrams under which the transformation from ungraded diagrams to graded versions is stable.
Speaker: Andrew Thomas
Title: Limit Theorems for Betti number and Euler characteristic processes
Abstract: In this study, we present limit theorems for Betti number and Euler characteristic processes of random ?ech and Vietoris-Rips complexes, where the points of the complexes are drawn from non-homogeneous Poisson processes in R^d. We focus on the critical (thermodynamic) regime, when the behavior of the radii of the balls governing the formation of the simplices leads to a highly connected complex with non-trivial homology. We prove that the Betti number process converges weakly in a finite-dimensional sense to a sum of centered Gaussian processes and that the Euler characteristic process converges weakly in the Skorohod space to a sum of centered Gaussian processes as well. Additionally, we establish a functional strong law of large numbers for the Euler characteristic process. These results complement tools such as the persistence diagram in topological data analysis (TDA) to capture the dynamic evolution of shape.
Speaker: Tomasz Kaczynski
Title: Towards an application driven extension of Forman's discrete Morse theory to multi-parameter functions.
Abstract: We propose new definitions of a vector-valued discrete Morse function, its gradient field, and its regular and critical cells. We present some properties analogous to the properties of a discrete Morse function established by Forman. This extension of the Forman’s theory is aimed at applying it to reduce complexes in the computation of multi-parameter persistent homology. We next show that multi-parameter data given on vertices on a simplical complex used as input for the reduction algorithm gives rise to a vector-valued discrete Morse function that preserves the partial matching produced by the algorithm. This talk is based on a joint work with Madjid Allili and Claudia Landi.
Speaker: Nicole Sanderson
Title: Topological Data Analysis for Time Series Using Witness Complexes
Abstract: Real-time regime shift detection between chaotic dynamical systems via time series analysis demands quick and correct, theoretically guaranteed methods. Often the best implemented techniques in the field are well-motivated heuristics and even interpretable statistics are scarce. Topological data analysis can contribute to the canon of traditional methods for analyzing nonlinear time series but is not computationally cheap. We introduce a family of topology-based classifiers that use a sparse complex - the witness complex - over a range of model parameter values affecting the efficiency of computations. We then explore the topology of witness complexes over this parameter range. We next define a simplicial complex whose construction incorporates the temporal information available with time series data. We experimentally show that this novel construction results in filtrations with fewer simplices and improved topological signature. We apply our techniques to synthetic time series data including numerical solutions of classical low dimensional chaotic systems Lorenz and R ̈ossler systems of ODEs as well as regimes of the higher dimensional Brunel neuronal network model and experimental live voltage recordings of musical instruments.
Speaker: Michael J Catanzaro
Title: Geometric perspectives on multiparameter persistence
Abstract: Inspired from geometric and differential topology, we introduce a version of multiparameter persistence combining sub-level and zig-zag persistence. Our construction arises from one-parameter families of smooth (Morse) functions on compact manifolds. We show how to analyze this version of multiparameter persistence in the language of geometric topology with several examples. Furthermore, we focus on practical aspects of this theory, with an emphasis on visualization and potential algorithm development.