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.