Das Forschungsseminar richtet sich an ein breites mathematisches Publikum. Es soll die Diskussionen über die mathematischen Spezialisierungen der verschiedenen Arbeitsgruppen hinaus am Institut fördern. Außerdem sollen auch Studierende (Master-Studierende und fortgeschrittene Bachelor-Studierende) durch das Seminar die Gelegenheit erhalten, sich über aktuelle Themen der Mathematik zu informieren.
28.07.2023, Ulmenstr. 69, Haus 3, HS 326/327
15:00 Uhr: Benno Rumpf (Southern Methodist University, Dallas, US)
Cold breathers: a statistical explanation for the localization of energy in nonlinear Klein-Gordon lattices
Localization of energy via the formation of discrete breathers is a common, but slightly counterintuitive event in the dynamics of nonlinear lattices. There is a simple statistical explanation for this phenomenon in lattices that conserve a second quantity in addition to the Hamiltonian; in the case of nonlinear Schrödinger systems this second quantity is the wave action. It turns out that the localization of energy is quite similar to the formation of droplets in saturated steam. Unfortunately, this explanation fails for lattices whose only conserved quantity is the Hamiltonian, e.g., nonlinear Klein-Gordon lattices. I will suggest a new explanation for this phenomenon.
26.07.2023, Ulmenstraße 69, Haus 3, HS 326/327
15:00 Uhr : Julia Slipantschuk (University of Warwick, Warwick, UK)
Transfer operators and applications
I will give a short introduction to the theory of transfer operators and explain how the spectral properties of these operators yield insight into the long-term behaviour of the underlying dynamical system. For sufficiently chaotic dynamical systems, the exponential mixing rates are obtained from the eigenvalues of the associated transfer operator. In very few non-trivial cases it is possible to determine all the mixing rates explicitly. On the other hand, it is often possible to approximate these operators by finite rank operators, and show convergence of the eigenvalues obtained from the respective matrix representations to the actual eigenvalues of the operator.
26.07.2023, Ulmenstr. 69, Haus 3, HS 326/327
16:00 Uhr: Oscar F. Bandtlow (Queen Mary University of London, London, UK)
Spectral approximation of evolution operators using dynamical mode decomposition
A class of algorithms known as extended dynamic mode decomposition (EDMD) has been shown to be empirically effective at identifying intrinsic modes of a dynamical system from time-series data. The algorithm amounts to constructing an NxN matrix by observing a dynamical system through N observables at a sequence of M phase space points. While empirically successful, there are few rigorous results on the convergence this algorithm. Moreover, the relationship between M and N remains obscure.
In this talk I will focus on analytic expanding circle maps and show that spectral data of the EDMD matrices can be linked to spectral data (for example eigenvalues, also known as Ruelle resonances in this context) of the Perron-Frobenius operator or its dual, the Koopman operator, associated to the underlying map, provided both operators are considered on suitable function spaces. In particular, I will show that for equidistantly chosen phase space points, spectra of the EDMD matrices converge to the Ruelle resonances at exponential speed in N, provided that the number of data points M is chosen to be a constant multiple of N, where the constant depends on complex expansion properties of the underlying analytic circle map.
This is joint work with Wolfram Just and Julia Slipantschuk.