Februar 2018

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### 01.02.2018 14:30 Dr. Carina Geldhauser (St. Peterburg) :Invariant measures of the Euler equation in vorticity form2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

When investigating the turbulent behaviour in a flow, it seems natural to study invariant measures of the Euler equation. In this talk we desribe the system of point vortices, derived by Onsager from the Euler equation, and their associated Gibbs measures. In case of constant circulations, Caglioti, Lions, Marchioro and Pulvirenti showed that the Gibbs measures concentrate on very particular stationary solutions of the 2D Euler equation in the weak limit and they satisfy a variational principle. Furthermore, we discuss available results and open problems for systems of point vortices with random circulations on the 2D Torus

### 05.02.2018 15:00 Annalisa Iuorio (TU Wien):Geometric Singular Perturbation Analysis of a Model for Micro-Electro Mechanical Systems (MEMS)MI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Many technological devices commonly used today rely on Micro-Electro Mechanical Systems (MEMS). These are deﬁned as very small structures combining electrical and mechanical components on a common substrate to perform several tasks. Their application in various ﬁelds such as medicine, transport industry and communications has raised considerable scientiﬁc interest.

The aim of this talk is to give a general introduction on the electrostatic-elastic case, where an elastic membrane is allowed to deﬂect above a ground plate under the action of an electric potential. This situation can be mathematically described by a parabolic PDE with a particular nonlinear source term that can lead to the “touchdown phenomenon”. Mathematically, touchdown causes non existence of steady states and/or ﬁnite time blow-up of solutions. A recently proposed model depending on a small “regularization” parameter $\varepsilon$ is introduced, where considering additional insulating effects allows to avoid singularities. We use tools from geometric singular perturbation theory and blow-up methods to study the bifurcation of steady-state solutions, emphasizing the interplay between the parameters appearing in the model. In particular, we focus our attention on the singular limit as these small parameters tend to zero.

### 05.02.2018 16:30 Dr. Christian Hirsch (LMU):Continuum percolation for Cox point processesB 252 (Theresienstr. 39, 80333 München)

We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

### 06.02.2018 14:15 Sergey Tikhomirov (Saint-Petersburg State University and John v. Neumann Professor @ TUM):Rattling in spatially discrete reaction-diffusion equations with hysteresisMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Hysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interacts according to a hysteresis law. Due to the discontinuity of hysteresis, these equations are not always well-posed.

We consider a spatial discretization of the problem and present a new mechanism of pattern formation, which we call rattling. The profile of the solution forms two hills propagating with non-constant velocity. The profile of hysteresis forms a highly oscillating quasiperiodic pattern, which explains mechanism of illposedness of the original problem and suggests a possible regularization. Rattling is very robust and persists in arbitrary dimension and in systems acting on different time scales.

The talk is based on joint works with P. Gurevich.

### 08.02.2018 14:30 Dominik Schmid:Mixing times for the exclusion process2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we focus on the exclusion process on finite graphs. Our goal is to study the total variation mixing time, which is the standard way of measuring the speed of convergence to equilibrium. We give an overview over some recent results on the mixing time of the exclusion process on the line segment as well as a selection of open problems in this area.

### 08.02.2018 16:30 Sabine Jansen:Effektive freie Energien, stochastische Grenzwertsätze und Metastabilität (Antrittsvorlesung)A 027 (Theresienstr. 39, 80333 München)

Effektive freie Energien für Ordnungsparameter in der klassischen statistischen Mechanik hängen eng mit stochastischen Grenzwertsätzen zusammen. Von besonderem Interesse sind nicht-konvexe freie Energien, da diese häufig mit dem interessanten dynamischen Phänomen der Metastabilität einhergehen: Markovprozesse, die ein vorgegebenes Gibbsmaß invariant lassen, brauchen möglicherweise lange Zeiten, um von einem metastabilen Gleichgewicht (man denke an lokale Minima der freien Energie) ein globales Gleichgewicht zu erreichen. Die Beantwortung der Frage "wie lang?" bedient sich mitunter Techniken der semi-klassischen Analysis. Der Vortrag konkretisiert einige dieser Zusammenhänge anhand des Widom-Rowlinson-Modells und beruht auf einem aktuellem Projekt mit F. den Hollander, R. Koteckÿ, and E. Pulvirenti.

### 12.02.2018 15:00 Prof. Dr. Sergey Tikhomirov (Universität Sankt Petersburg):Shadowing of infinite and finite pseudotrajectoriesMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Starting from celebrated Anosov shadowing lemma relations between hyperbolicity, structural stability and shadowing has been studied in a lot of works. Most of the results are proving that hyperbolicity/structural stability implies shadowing. In this talk we would mostly interested in the conversed direction. We study question: under which conditions shadowing implies hyperbolicity/structural stability?

On the possible approaches goes back to Sakai, where he proved that C^1-robust shadowing for diffeomorphisms is equivalent to structural stability. Another approach is related to quantitative characteristics of shadowing. It follows from original prove by Anosov that for hyperbolic sets shadowing is in fact Lipschitz. A similar results hold for structurally stable flows.

In this overview of results obtained in 2008 – 2017 we consider both approaches and show that Lipschitz shadowing is equivalent to structural stability for diffeomorphisms and flows. Generalization of results on C^1-robust shadowing to the case of flows is not so straightforward. We show that there exists a not structurally stable flow with C^1 -robust shadowing and prove that in some sense it is the only possible example.

### 15.02.2018 14:00 Greg Henselman (Princeton University):The Matroid of Barcodes: Combinatorial Foundations in TDAMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Topological data analysis (TDA) is a robust field of mathematical data science specializing in complex, noisy, and high-dimensional data. While the elements of modern TDA have existed since the mid-1980’s, applications over the past decade have seen a dramatic increase in systems analysis, engineering, medicine, and the sciences. Two of the primary challenges in this field regard modeling and computation: what do topological features mean, and are they computable? While these questions remain open for some of the simplest structures considered in TDA — homological persistence modules and their indecomposable submodules — in the past two decades researchers have made great progress in algorithms, modeling, and mathematical foundations through diverse connections with other fields of mathematics. This talk will give a first perspective on the idea of matroid theory as a framework for unifying and relating some of these seemingly disparate connections (e.g. with quiver theory, classification, and algebraic stability), and some questions that the fields of matroid theory and TDA may mutually pose to one another. No expertise in homological persistence or general matroid theory will be assumed, though prior exposure to the definition of a matroid and/or persistence module may be helpful.

### 20.02.2018 16:15 Paul Carter (University of Arizona):Canard induced pulse-adding and spike-adding transitions in slow-fast systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We focus on the FitzHugh--Nagumo system, a simplified model of nerve impulse propagation, which is known to exhibit stable traveling pulse solutions. Upon parameter continuation, these pulse solutions may undergo a sharp transition, reemerging as double pulse solutions. We describe a construction of this transition from single to double pulses using geometric singular perturbation theory, and we demonstrate the integral role of canards, which arise as tangencies of repelling and attracting slow manifolds. We also discuss ongoing work involving similar geometric constructions in neuronal bursting models, which exhibit continuous spike-adding transitions between periodic bursting solutions with different numbers of spikes.

### 20.02.2018 16:30 Camille Male (Université de Bordeaux - Institut de Mathématiques de Bordeaux):Distributional symmetries and non commutative independencesMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

The properties of the limiting non commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e.g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. Nevertheless, the study of random matrices invariant in law by conjugation by permutation matrices requires an extension of free probability, which motivated the speaker to introduce in 2011 the theory of traffics. A traffic is a non commutative random variable in a space with more structure than a general non commutative probability space, so that the notion of traffic distribution is richer than the notion of non commutative distribution. It comes with a notion of independence which is able to encode the different notions of non commutative independence.

### 27.02.2018 11:00 Maik Kahnt (DESY):2D and 3D ptychographyMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

X-ray ptychography is the newest and most promising microscopy technique at synchrotron sources. The resolution is neither limited by fabrication errors of the optics, the size of the sample nor the beam size on the sample. This talk focuses on the ptychographic phase retrieval algorithm and the challenges when it is applied to real experimental data.

### 28.02.2018 16:15 Dr. Peter Giesl (Department of Mathematics, University of Sussex, UK):Existence and construction of a contraction metric as solution of a matrix-valued PDEMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.

A contraction metric can be used to prove existence and uniqueness of an equilibrium or periodic orbit of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system.

We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium and a periodic orbit, respectively, by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones.

This is partly work with Holger Wendland, Bayreuth.