Abstracts and resources

In this mini-course we will explore connections between algebraic and analytic positivity. Starting with polynomial rings we will show how to use the solution to Hilbert 17-th problem to approximate witnesses of positivity using semi-definite optimization. Such witnesses take form of sum of squares (of rational functions), which form a hierarchy of semi-definite optimization problems which we can solve efficiently. Unfortunately such formulations tend to grow very quickly and soon become computationally intractable. Thus we turn our attention to symmetries of those rings which can provide an elegant way of reducing (via Wedderburn-Artin decomposition) the size and complexity of the optimization problems. Such symmetry reduction may lead to instabilities when performed directly using linear algebra and floating point arithmetic. We will explore ways of using symbolic algebra to avoid those numerical problems and delay the expensive linear algebra computations for as long as possible. Finally we will showcase how to use those methods to bound norm of an operator. If time permits we will showcase two applications: one in quantum information and one in geometric group theory.

slides

In this lecture we will present a basic use-cases of techniques of TDA, in particular persistent homology and mapper algorithm. We will provide both theoretical exposition and show how to use the theory in practice on a computer.

slides

This lecture will be given from the perspective of a statistician on how topological representations of data can allow for the analysis of complex data using either existing statistical models or simple adaptations. Also how topological summaries can provide value added to statistical models. Most of the case study examples will involve shape but some other examples will be given.

R demo

slides

Reinforcement learning has witnessed spectacular growth due to the application of deep learning. RL addresses the problem of sequential decision making by studying credit assignment of the future value on the current action. It is a very flexible description allowing for tackling many practical problems. In this short course, I will preset the motivation of RL, its formalism, and fundamental algorithms. If time permits, I’ll make a short practical demonstration in colab (please bring laptops with you).

Discrete Morse theory is a tool that investigates both topological and combinatorial properties of a simplicial complex in order to determine homotopy type, estimate or compute Betti numbers, and discern other information about the complex. In this workshop, we will study discrete Morse theory on simplicial complexes by discussing both the basics of discrete Morse theory and giving a broad overview of some of the recent and more advanced techniques that have been developed. We will study discrete Morse theory from the three perspectives of gradient vector fields, discrete Morse functions, and acyclic matchings on the Hasse diagram before working through the classical homotopy results of Forman. We will then give a broad overview of some of the variations of discrete Morse theory including Bestvina-Brady Morse theory, discrete stratified Morse theory, and using discrete Morse theory to explicitly compute homology cycles. This workshop should be accessible to anyone with a basic understanding of simplicial complexes and some familiarity with the idea of homology.

Handout

slides

In this mini-course, a method for automated analysis of dynamics in a semi-dynamical system that depends on a few parameters will be explained. This method was introduced in [1] and was successfully applied to some discrete-time dynamical systems and to flows as well; the applications range from population models [2] through theoretical physics [3] to epidemiology [4].

The proposed approach uses rigorous numerical methods and topological methods based on the Conley index and Morse decompositions, and provides mathematically validated results concerning the qualitative dynamics of the system. The main idea is to split the phase space (a subset of Rn ) into a rectangular grid, and to use interval arithmetic to compute an outer estimate of the map on the grid elements. This construction gives rise to a directed graph, and fast graph algorithms allow to enclose all the recurrent dynamics in bounded subsets so that the dynamics outside the collection of these sets is gradient-like, and thus a discrete version of a Morse decomposition is constructed. The homological Conley index is computed for each of the discrete Morse sets in order to determine its stability and other features. A splitting is computed for the entire range of parameters under consideration into classes that yield equivalent dynamics. All these computations are carried out automatically by efficient software written in C++. As a result, an interactive Web page is created for browsing the results, as shown in [5].

In addition to explaining the method and the results it provides, the mini-course also contains hands-on experience with the software for conducting the computations, so that the participants should be able to apply this software to the dynamical systems of their interest.

[1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk. A database schema for the analysis of global dynamics of multiparameter systems. SIAM J. Appl. Dyn. Syst., Vol. 8, No. 3 (2009), 757–789. DOI: 10.1137/080734935.

[2] E. Liz, P. Pilarczyk. Global dynamics in a stage-structured discrete-time population model with harvesting. J. Theoret. Biol., Vol. 297 (2012), 148–165. DOI: 10.1016/j.jtbi.2011.12.012.

[3] P. Pilarczyk, L. García, B.A. Carreras, I. Llerena. A dynamical model for plasma confinement transitions. J. Phys. A: Math. Theor., Vol. 45, No. 12 (2012), 125502. DOI: 10.1088/1751-8113/45/12/125502.

[4] D.H. Knipl, P. Pilarczyk, G. Röst. Rich bifurcation structure in a two-patch vaccination model. SIAM J. Appl. Dyn. Syst., Vol. 14, No. 2 (2015), 980–1017. DOI: 10.1137/140993934.

[5] P. Pilarczyk. Databases for the global dynamics of multiparameter systems. http://www.pawelpilarczyk.com/database/ (accessed on April 12, 2022)