In this talk, I consider a Swift-Hohenberg equation which is coupled to a conservation law. As a bifurcation parameter increases beyond a critical value this system undergoes a Turing bifurcation and small, spatially periodic solutions emerge from a homogeneous ground state. To model the transition of the ground state to the periodic state, the existence of modulating traveling front solutions is established. These fronts model an invasion of the ground state by the periodic state. I outline the proof, which is based on spatial dynamics and center manifold reduction, and discuss new challenges, which arise from an additional neutral mode at Fourier wave number $k=0$.

Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. We introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape [1]. We construct a family of conditionally-invariant measures with varying decay rates by interpo- lating between the natural measures of the forward and backward dynamics. We show numerically, that these classical measures describe the main features of quantum resonance eigenfunctions: their multifractal phase space distribution, their product structure along stable and unstable directions, and their dependence on the decay rate. The (Jensen-Shannon) distance between classical and quantum measures goes to zero in the semiclassical limit for long- and short-lived eigenfunctions, while it remains finite for intermediate cases.

[1] K. Clauß, E. G. Altmann, A. Bäcker, and R. Ketzmerick, Phys. Rev. E 100 (2019), 052205.

Nonlinear PDEs are widely used in several and diverse fields of Physics such as statistical physics, biophysics, and fluid mechanics. A particular type of nonlinear PDEs are especially important for quantum mechanics, the nonlinear Schroedinger equation (NLSE). Far from being only of theoretical interest, they are useful in applied physics and practical problems. They describe wave diffraction in the ionosphere, light propagation in optical fibers, Bose-Einstein condensates, nuclear resonance, and others. I am going to explore the solutions of the NRT and Plastino-Rocca equations and their approximations for the weakly nonlinear limit. Those solutions describe exponentially decaying resonance states, the q-Gamow states. The associated q-Breit-Wigner distribution is computed, which is, in principle, amenable to empirical verification.

In this talk, I will approach the topic of bifurcations from two different perspectives. At first, we will focus on validated proofs of bifurcation, specifically saddle nodes and Hopf bifurcations of periodic orbits in ODEs. We will apply a blow-up technique to desingularise a Hopf bifurcation and prove its existence with rigorous numerics. To showcase the generality of this method, I will present its application in ODEs and in (one) PDE. Then, I will talk about bifurcations in cell biology. These biological models are notorious for having a very high dimensional parameter space. In such setting, bifurcations are plentiful, but experiments only exhibit few behaviors and parameters are hard to determine. We want to determine which parameter region is more likely to exhibit the dynamical behavior shown in experiments. We do this by studying the bifurcations of the system.

In this talk we analyse the long-time behaviour of solutions to two linear kinetic PDEs with defects: The degenerate Fokker--Planck equation and the Goldstein--Taylor system (a two velocity transport-relaxation model of BGK-type). Both equations model the evolution in time of an average particle of a large system of particles under the influence of two forces: one conservative and one dissipative. Our discussion focuses on the defective cases that occur in these models, which, much like finite dimensional defective ODEs, imply a polynomial times exponential convergence of solutions. To obtain explicit estimates, we construct tools for entropy methods and utilise spectral theory in a non-symmetric setting.

So please find here the invitation details:

Topic: Oberseminar Finanz- und Versicherungsmathematik Time: Oct 22 2020 10:15 AM Bruxelles

Join Zoom Meeting https://lmu-munich.zoom.us/j/9328461914?pwd=b2plMkwyVFNrMWxwZktZMlBlM0Y4Zz09

Meeting ID:: 932 846 1914 Passcode: 148394

More information about the talk at the following link:

https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_winter_term_2020_2021/seminars/oberseminar_finanz_2019/index.html

Starting with a general introduction to statistical relational artificial intelligence (StarAI), I will continue to show how analysing StarAI frameworks under the lens of first-order logics of probability can help us to approach two interlinked problems, which have recently garnered much attention in the field of statistical relational AI:

(1) How can one apply statistical relational methods on large domains given the high complexity of learning and inference?

(2) How can parameters learned on one domain be transferred to a domain of different size?

Meeting ID: 925-6562-2309, Password on demand: Office.Leitgeb@lrz.uni-muenchen.de