Juni 2020

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### 08.06.2020 15:00 Kateryna Marynets (TU Delft):Stuart–type vortices modeling the steady flow of the Antarctic Circumpolar CurrentVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

Using the method of sub- and supersolutions for semilinear elliptic equations, in combination with the stereographic projection, we show that Stuart-type vortices can model the steady flow of the Antarctic Circumpolar Current. Further analysis is provided.

### 08.06.2020 16:00 Marek Biskup (UCLA):A quenched invariance principle for random walks with long range jumps(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

I will discuss random walks among random conductances on the hypercubic lattice that allow for jumps of arbitrary length. This includes the random walk on the long-range percolation graph obtained by adding to $\mathbb Z^d$ an edge between $x$ and $y$ with probability proportional to $|x-y|^{-s}$, independently of other pairs of vertices. By a combination of functional inequalities and location-dependent truncations, I will prove that the random walk scales to Brownian motion under a diffusive scaling of space and time. The proof follows the usual route of reducing the statement to everywhere sublinearity of the corrector. We prove the latter under moment conditions on the environment that in fact turn out to be more or less necessary for the method of proof. For the above percolation problem, this requires the exponent~$s$ to exceed~$2d$. Based on joint work with X. Chen, T. Kumagai and J. Wang.

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### 15.06.2020 14:15 Prof. Thorsten Schmidt :tba onlineMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

Due to the current situation, the Oberseminar of 15.06.20 will be held online on Zoom. People interested in attending the seminar are invited to write an e-mail to mazzon@math.lmu.de in order to receive an invitation to the meeting.

https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_summer_term_2020/seminars/oberseminar_finanz_2019/index.html

### 15.06.2020 15:00 Noémie Ehstand (IFISC /IUB-CSIC), Spain):Numerical continuation of fractional PDEs: sharp teeth and bloated snakesVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

Partial differential equations (PDEs) involving fractional Laplace operators have attracted increasing research attention as they have proven useful for modeling anomalous diffusion processes in areas as varied as physics, biology, ecology, medicine and economics. In this talk, we explore the effects of the spectral fractional Laplacian on the solutions and bifurcation structure of fractional reaction–diffusion systems on bounded domains. We first extended the continuation/bifurcation package pde2path, which has been extensively used for classical reaction–diffusion systems, to treat PDEs involving the spectral fractional Laplacian. The new capabilities of the software were then applied to the study of three benchmark problems: the Allen–Cahn equation, the Swift–Hohenberg equation and the Schnakenberg system in which the standard Laplacian was replaced by the spectral fractional Laplacian. Our results have shown that the fractional order induces significant qualitative and quantitative changes in global bifurcation structures, of which some are shared by the three systems. This contributes to a better understanding of the effects of fractional diffusion in generic reaction–diffusion systems. (Presentation is based on a joint work with Christian Kuehn and Cinzia Soresina.)

### 15.06.2020 16:00 Franziska Kühn (TU Dresden):Regularity theory for non-local operators(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

Let $A$ be the infinitesimal generator of a Lévy process. Classical examples are, for instance, the Laplacian (generator of Brownian motion) and the fractional Laplacian (generator of isotropic stable Lévy process). In this talk, we study the regularity of solutions $f$ to the Poisson equation $Af=g$. We show how gradient estimates for the transition density of the Lévy process can be used to obtain Hölder estimates for $f$. Moreover, we present a Liouville theorem for Lévy operators: If $f$ is a solution to $Af=0$ which is at most of (suitable) polynomial growth, then $f$ is a polynomial. We illustrate our results with examples and discuss some possible generalizations.

t.b.a.

### 22.06.2020 15:00 Andreas Burkhart:Moment equations and moment-closure approximations of the Adaptive Voter Model and Adaptive Simplex Voter Model Virtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

A small talk about adaptive and non-adaptive networks and the development of their respective moment equations/moment-closure approximations to represent the long term systemic development. In particular I will shortly cover the Adaptive Voter Model and Adaptive Simplex Voter Model as an example from my master's thesis.

### 22.06.2020 16:00 Sabine Jansen (LMU):Phase transitions for a hierarchical mixture of cubes(using zoom) (Theresienstr. 39, 80333 München)

We consider a discrete toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Z^d with side-lengths 2^j, j\in N_0. Cubes belong to an admissible set B such that if two cubes overlap, then one is contained in the other, a picture reminiscent of Mandelbrot's fractal percolation model. I will present exact formulas for the entropy, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and explain how the toy model fits into a renormalization program for mixtures of hard spheres in R^d. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

TBA