# Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45, Bonn

Organizers: Herbert Edelsbrunner, Kathryn Hess, Michael Farber, Dmitry Feichtner-Kozlov, Martin Raussen

## Wednesday, September 6

14:30 Gard Spreemann: Using persistent homology to study covariates in neuronal and other systems modeled by the kinetic Ising model

## Thursday, September 7

14:30 Ran Levi: Neuro-topology: An interaction between topology and neuroscience

## Wednesday, September 13

10:30 Dmitry Feichtner-Kozlov: Simplicial combinatorics of weak symmetry breaking

14:30 Krzysztof Ziemianski: Directed path spaces via discrete vector fields

## Thursday, September 14

10:30 Éric Goubault: On a notion of directed homotopy equivalence and topological complexity

11:40 Jérémy Ledent: Standard chromatic subdivision via partial cube paths

## Friday, September 15

10:30 Nicolai Vorobjov: Approximation of definable sets by compact families

Abstract: In real analytic geometry (and, of course, more generally) compact definable sets are easier to handle than arbitrary ones. I will describe a constructive method of compactification of semialgebraic or subanalytic sets, based on their approximations by compact families, and its application to upper bounds on Betti numbers.

## Monday, September 18

14:30 Vanessa Robins: Topological crystallography

## Tuesday, September 19

14:30 Wojtek Chacholski: What is persistence?

## Wednesday, September 20

14:30 Brittany Fasy: Confidence in persistent homology

## Thursday, September 21

14:30 Frank Lutz: Topology of steel

## Monday, September 25

14:30 Dima Grigoriev: Bounds on the topology of tropical prevarieties

Abstract: A digest on the basics of tropical geometry with focus on effectiveness:

- historical sources and applications;
- tropical varieties and prevarieties;
- tropical linear algebra;
- min-plus prevarieties and game theory;
- tropical Nullstellensatz;
- bounds on the number of connected components of a tropical prevariety.

## Tuesday, September 26

14:30 Primoz Skraba: Minimum spanning acycles and persistence

Abstract: Minimum spanning acycles are a natural generalization of minimum spanning trees for simplicial complexes. Building on the work of Hiraoka and Shirai, I will describe joint work with Yogeshwaran D. and Gugan Thoppe on understanding these objects. I will discuss several connections with persistent homology and how stability can help us analyze the properties of minimum spanning acycles of random complexes.

## Wednesday, September 27

14:30 Gaiane Panina: Diagonal complexes

Abstract: Generalising a construction of J.L. Harer we introduce and study diagonal complexes related to a (possibly bordered) oriented surface F equipped with a number of labeled fixed points. Investigation of some natural forgetful maps combined with length assignment proves homotopy equivalence of some of the complexes to the space of metric ribbon graphs RG^{met}_{g,n}, to the (introduced by M. Kontsevich) tautological S^{1}-bundles L_{i}, to the Whitney sum of L_{i}, and to a more sophisticated bundle whose fibers are homeomorphic to some surgery of the surface F. As an application, we compute the powers of the first Chern class of L_{i} in combinatorial terms. The latter result is an application of N. Mnev and G. Sharygin local combinatorial formula.

This is a joint with Joseph Gordon, St. Petersburg State University.

## Thursday, September 28

14:30 Marian Mrozek: Persistence of Morse decompositions

Abstract: Morse decompositions capture the gradient structure of a flow. This structure may change when a parameter varies. I'll present some ideas how to measure persistence of these changes for combinatorial vector fields of Forman and their multivector generalizations. This is research in progress, joint with T. Dey, M. Juda, T. Kapela and J. Kubica.