Das Mathematische Kolloquium ist eine gemeinsame wissenschaftliche Veranstaltung des gesamten Mathematischen Instituts. Es steht allen Interessierten offen und richtet sich neben den Mitgliedern und Mitarbeitern des Instituts auch an die Studierenden. Das Kolloquium findet dreimal im Semester am Donnerstag um 15:00 s.t. im Hörsaal II, Albertstr. 23b statt. Danach (gegen 16:15) gibt es Kaffee und Kekse, zu dem der vortragende Gast und alle Besucher eingeladen sind.
Magnetic skyrmions
Donnerstag, 2.5.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
Donnerstag, 6.6.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
Analysis of Correlated Many-Body Systems: Bose-Einstein Condensates and Mean Field Spin Glasses
Donnerstag, 13.6.24, 08:30-09:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, I provide an overview of my recent research on the mathematical analysis of correlated many-body systems. I describe exemplary results concerning dilute quantum systems of interacting bosons and concerning disordered mean field systems of interacting Ising spins. In both cases, I carefully introduce the model, review relevant results and outline current as well as future research directions. The talk is based on joint work with M. Brooks, C. Caraci, J. Oldenburg, A. Schertzer, C. Xu and H.-T. Yau. .
Stabilization by transport noise and enhanced dissipation in the Kraichnan model
Donnerstag, 13.6.24, 10:40-11:40, Raum 404, Ernst-Zermelo-Str. 1
Thanks to the work of Arnold, Crauel, and Wihstutz it is known that for any self-\nadjoint operator T acting on a finite dimensional space with the negative trace the\ncorresponding linear equation dxt = T xt dt can be stabilized by a noise, i.e. there\nexists operator-valued Brownian motion W such that the solution of dxt + dW xt =\nT xt dt vanishes a.s. for any initial value x0 = x. The goal of the talk is to extend this\ntheorem to infinite dimensions. Namely, we prove that the equation dut = ∆ut dt\ncan be noise stabilized and that an arbitrary large exponential rate of decay can\nbe reached. The sufficient conditions on the noise are shown to be satisfied by the\nso-called Kraichnan model for stochastic transport of passive scalars in turbulent\nfluids. This talk is based on joint work with Prof. Benjamin Gess (MPI MiS and\nBielefeld University).
Sub-Riemannian geometries and hypoelliptic diffusion processes
Donnerstag, 13.6.24, 14:00-15:00, Raum 404, Ernst-Zermelo-Str. 1
I will start with an overview on sub-Riemannian geometries, where motion is only possible along certain admissible trajectories, and hypoelliptic diffusion processes, which due to underlying constraints spread in different directions at different orders. Subsequently, I will present two projects where the analysis of stochastic processes on constrained systems has proven to be fruitful. Firstly, I will discuss how a stochastic process introduced jointly with Barilari, Boscain and Cannarsa on surfaces in three-dimensional contact sub-Riemannian manifolds can be used to classify singular points arising in that setting. Secondly, I will show how the study of a standard one-dimensional Brownian motion conditioned to have vanishing iterated time integrals of all orders, which can be rephrased as studying projected hypoelliptic diffusion loops, has led to a novel polynomial approximation for Brownian motion..
Singular PDEs: regularity and homogenization
Donnerstag, 13.6.24, 16:10-17:10, Raum 404, Ernst-Zermelo-Str. 1
I will mostly focus on the regularity theory and homogenization for elliptic equations with degenerate unbounded coefficients, both at the deterministic (mostly done with Mathias Schäffner) as well as stochastic level. While already interesting on its own, I will mention two areas of use for these: study of regularity of critical points for variational integrals as well as invariance principle for random walks in random environments. At the end, I will conclude with a short discussion of few result in quantitative stochastic homogenization..
A semigroup approach for stochastic quasilinear equations driven by rough noise
Freitag, 14.6.24, 08:00-09:00, Raum 404, Ernst-Zermelo-Str. 1
We consider stochastic quasilinear equations perturbed by nonlinear multiplicative noise. Ex-ploring semigroup methods and combining techniques from functional analysis with tools from rough path theory, we establish the pathwise well-posedness of such equations. We apply our results to the stochastic Shigesada-Kawasaki-Teramoto equation describing population segregation by induced cross-diffusion and to the Landau-Lifshitz-Gilbert equation which models the magnetization of a ferromagnetic material. Moreover, we emphasize the advantage of rough path theory in the study of the long-time behavior of such systems. This talk is based on joint works with Antoine Hocquet and Christian Kuehn.
Optimal Transport and Diffusion on varying spaces
Freitag, 14.6.24, 10:10-11:10, Raum 404, Ernst-Zermelo-Str. 1
We discuss contraction estimates of diffusion under optimal transport problems on varying spaces. We further investigate in equivalent formulations and generalizations of these estimates.
From microscopic to macroscopic scales: effective evolution equations of many interacting particles.
Freitag, 14.6.24, 13:30-14:30, Raum 404, Ernst-Zermelo-Str. 1
Systems of interacting particles describing notable physical phenomena, such as time-irreversibility, Bose-Einstein condensation, superconductivity or superfluidity, represent a veritable challenge for mathematicians and physicists. They exhibit a daunting complexity, which renders the exact many-body theory non-approachable, not only from a mathematical viewpoint, but also for computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics. In this talk, I will present several settings leading to different effective kinetic equations and then I will focus on the mean-field regime for quantum particle systems, highlighting recent significant progress in the mathematical understanding of these systems.
Oscillatory effects in stochastic PDEs
Freitag, 14.6.24, 15:40-16:40, Raum 404, Ernst-Zermelo-Str. 1
I will describe two recent results related to the oscillations of the noise in stochastic PDEs. In the first example, a rougher-than-usual KPZ equation, the fluctuation of the noise is strong enough that on the relevant scale the nonlinearity turns into a new Gaussian noise via a central limit-type theorem. The second example, a 1-dimensional stochastic Allen Cahn equation, is much less singular and a solution theory is fairly unproblematic. Nevertheless, the averaging effects of the noise play a key role in their discretisations: they exhibit four times better temporal pointwise convergence rate than the pointwise regularity of the solution and twice better than the regularity of its single Fourier mode. Based on joint works with Ana Djurdjevac, Helena Kremp, Fabio Toninelli.
Der angewandte Mathematiker Henry Görtler vor und nach 1945.
Donnerstag, 27.6.24, 15:00-16:00, Hörsaal II, Albertstr. 23b