Goście i wykłady

Herbert Edelsbrunner (Institute of Science and Technology Austria)

To be announced
Abstrakt TBA

Więcej informacji o naszym gościu: https://pub.ista.ac.at/~edels/

Thomas Wanner (George Mason University)

Combinatorial Topological Dynamics
Abstrakt 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.

Więcej informacji o naszym gościu: https://math.gmu.edu/~wanner/

Bogdan Batko (Jagiellonian University)

Więcej informacji o naszym gościu: https://matinf.uj.edu.pl/pracownicy/wizytowka?person_id=Bogdan_Batko

Paweł Dłotko (Dioscuri Centre in Topological Data Analysis)

Topological Data Analysis and beyond
Abstrakt 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.

Więcej informacji o naszym gościu: https://dioscuri-tda.org/members/pawel.html

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

Persistent homology in combinatorial dynamical systems
Abstrakt 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.

Więcej informacji o naszym gościu: https://sites.google.com/view/michal-lipinski/home