Beomjun Sohn (RWTH Aachen)
Topological entropy under small area perturbation abstract
Abstract:
In this talk, we present a new robustness result for the topological entropy of Hamiltonian diffeomorphisms on closed surfaces. Topological entropy is a fundamental measure of orbital complexity in dynamical systems, capturing chaotic behavior through a single non-negative value. We prove that if a Hamiltonian diffeomorphism has positive entropy, then any perturbation supported in a topological disk of sufficiently small area still has positive entropy. The proof relies on a new braid stability result derived from Floer-theoretic braids of fixed points, which can be related to the area of the support. This talk is based on ongiong joint work with Marcelo Alves and Matthias Meiwes.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Erika Roldan
Title T.B.A.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Stefan Steinerberger (UW Seattle)
Particle Dynamics and Dimensionality Reduction abstract
Abstract:
Dimensionality reduction, moving data from very high dimensions to intermediate dimensions, is well-established. It is not too difficult to accurately map n points into ~log(n) dimensions. The problem becomes a lot more difficult if one insists that the high-dimensional data be embedded into 2 dimensions which is what researchers in the biomedical fields need. Moreover, there are algorithms (most prominently tSNE and UMAP) that "semi-successfully" do this: they tend to work (though they also tend to fail on very simple examples). An added difficulty is that it is difficult to assess from the output when they work. I will argue that these algorithms reduce to understanding an attraction-repulsion functional acting on a large number of particles; this viewpoint immediately suggests new ideas and algorithms as well as a few benchmark problems.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room F 3
Prof. Dr. Karl Worthmann (TU Illmenau)
Abstract:
Extended dynamic mode decomposition (EDMD), embedded in the Koopman framework, is a widely-applied data-driven technique to predict the evolution of observables along the flow of a nonlinear dynamical system. However, despite its popularity, the error analysis is still fragmentary. First, we provide an analysis of the error resulting from a finite dictionary size (projection) and finite data (estimation). Second, we indicate extensions towards reproducible kernel Hilbert spaces to establish L∞-error bounds using kernel EDMD. Then, we leverage our findings to derive stability guarantees in predictive control despite using data-driven surrogate models in the optimization step.
17:15 • Universität Bern, Sidlerstrasse 5, 3012 Bern, Room B6