Guests and lectures

Herbert Edelsbrunner (Institute of Science and Technology Austria)

Chromatic persistent homology
Abstract Motivated by advances in spatial biology, we introduce the chromatic Delaunay mosaic of s+1 finite sets in d dimensions, which is an (s+d)-dimensional Delaunay mosaic that represents the individual sets as well as their interactions.
It supports a generalized discrete Morse radius function whose sublevel sets are the chromatic alpha complexes.
We prove bounds on the size of the chromatic Delaunay mosaic, in the worst and average cases, and introduce the 6-pack of persistence diagrams to stable descriptions of the interactions between the different colors.
This is joint work with Sebastiano Cultrera, Ondrej Draganov, and Morteza Saghafian, all at IST Austria.

More information about our guest: https://pub.ista.ac.at/~edels/

Thomas Wanner (George Mason University)

Combinatorial Topological Dynamics
Abstract Topological methods have always been at the heart of the qualitative study of dynamical systems. For example, topological fixed point theorems can guarantee the existence of stationary states, and classical Morse theory establishes a link between gradient dynamics and the topology of the underlying phase space. These ideas have been generalized to Conley's theory of isolated invariant sets and Morse decompositions, which is able to encode the global dynamics of general dynamical systems using topological information. In fact, even more complicated behavior can be studied in this way, for example recurrent and chaotic dynamics.

In recent years, some of the above approaches have been carried over to a discrete, combinatorial setting, thereby opening the door to a purely combinatorial version of dynamical systems. The first step in this direction was Forman's discrete Morse theory on simplicial complexes, and it has found numerous applications in mathematics, computer science, and the applied sciences.

In these lectures, I will present the current state of combinatorial topological dynamics, which generalizes and extends these early results in a number of directions. The notion of multivector fields on Lefschetz complexes will be introduced in detail, as well as an extension of Conley's theory in this setting. This includes topics such as isolated invariant sets, their Conley index, and Morse decompositions. In addition, we address applications of the theory to establishing recurrence in classical dynamics, and describe advanced topics such as connection matrices. Finally, we also touch upon the relation between this combinatorial theory and its classical counterpart.

More information about our guest: https://math.gmu.edu/~wanner/

Bogdan Batko (Jagiellonian University)

Conley index methods in data-driven dynamics
Abstract Conceptual models for most physical systems are based on a continuum; values of the states of a system are assumed to be real numbers.
At the same time science is increasingly becoming data driven and thus based on finite information. Traditional, analytical analysis of dynamical systems is possible only when the mathematical model of the dynamics is available. Frequently, in particular in experimental sciences, in biological, in sociological sciences or in medicine, this is rarely the case. Instead, scientists collect data by experiments. This suggests the need for tools that seamlessly and systematically provide information about continuous structures from finite data and accounts for the rapid rise in use of methods from topological data analysis. In this context sampled dynamics attracts interest of scientists. Not surprisingly, however, there are significant challenges associated with understanding of the data dynamics. So far, very little has been achieved in this regard.

The aim of these lectures is to discuss basics of the Conley index theory for multivalued maps and its applications to the study of dynamical systems known only from data. To this end we will introduce several concepts, including: solutions, isolated invariant sets, weak index pairs. Moreover, we will briefly discuss properties of the index, and Morse decompositions. We will also tackle Gaussian process surrogate modeling in sampled dynamics.

More information about our guest: https://matinf.uj.edu.pl/pracownicy/wizytowka?person_id=Bogdan_Batko

Paweł Dłotko (Dioscuri Centre in Topological Data Analysis)

Topological Data Analysis and beyond
Abstract Topological data analysis provides several methods to quantify the shape of data and visualize high-dimensional datasets. These methodologies have made a significant impact on various branches of mathematics and physics, as well as biological, medical, and political sciences, and beyond. In this lecture, I will provide a gentle, bird's-eye introduction to the methodology of topological data analysis. We will start with classical concepts of persistent homology and move to its variations, including Euler characteristic curves and profiles. We will show how these ideas can be used in different branches of mathematics, including statistical hypothesis testing. Subsequently, we will explore topological visualization methods, discussing how ideas from algebraic topology allow us to understand the layout of high-dimensional samples of points. We will conclude the lecture with an introduction to Forman's discrete Morse theory. All topics will be discussed from both mathematical and algorithmic perspectives, and some software demonstrations will be provided. Strong emphasis will be put on the used cases of the presented methodology.

More information about our guest: https://dioscuri-tda.org/members/pawel.html

Michał Lipiński (Jagiellonian University, Institute of Science and Technology Austria)

Persistent homology in combinatorial dynamical systems
Abstract In recent years, interest in the theory of combinatorial dynamical systems has been gaining momentum. Modern approaches are largely inspired by Forman's combinatorial vector fields, which are based on discrete Morse theory. This theory has found numerous applications in computational topology and topological data analysis. One extension of Forman's idea is the theory of multivector fields, which is developing into a robust theory on its own.

Simultaneously, the theory of persistent homology has flourished as a central concept in topological data analysis. The combinatorial framework of multivector fields aligns naturally with persistent homology, and thus, persistence can be utilized to understand dynamics.

In these lectures, we will discuss some recent results and ideas that link persistent homology and multivector fields, revealing interesting interplays between topology and dynamics.

More information about our guest: https://sites.google.com/view/michal-lipinski/home