November 2017

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### 02.11.2017 16:30 Dirk Deckert:N-body Problem of Classical Electrodynamics (Habilitationsvorstellung)A 027 (Theresienstr. 39, 80333 München)

I will report on a series of results on the solution theory of the coupled Maxwell and Lorentz equations describing the motion of N classical charges. I will start the discussion with a system of N rigid charges whose solutions can be obtained in terms of an initial value problem and explain the major obstacles in passing to the physically desirable point charge limit. In particular, I will demonstrate that already the Maxwell-Lorentz equations for point charges without self-interaction do not admit an initial value problem in the common sense as generic initial values lead to irregularities in the electromagnetic fields that travel along the light-cones of the initial charge positions and prevent global existence. Global smooth solutions can only be obtained by imposing an infinite system of constraints which effectively turn the Maxwell-Lorentz system into a system of delay differential equations. This system of delay equations is neutral, non-linear and involves state-dependent delays which renders its mathematical study very cumbersome. I will close with a discussion of recent results concerning existence and uniqueness of its solutions.

### 06.11.2017 10:00 Wojciech De Roeck (KU Leuven):Quantization of Hall conductance and a glimpse of 'edge modes' in interacting systems00.09.038 (Boltzmannstr. 3, 85748 Garching)

We review the question why conductance is quantized in interacting gapped systems. We present a streamlined version of the proof by Hastings, Michalakis (2009) covering both the integer and the fractional case. Then, we consider a system that, far to the right and far to the left, has a different value of the Hall conductance and we prove that such a system does not admit a gap. This is hence a rigorous version of a well-known bulk-edge principle, but applying just as well in cases where the spatial transition region is not a smooth interpolation between left and right system. Finally, we prove from first principles the validity of the linear response formalism that underlies the above discussion. This is joint work with Sven Bachmann, Alex Bols and Martin Fraas.

### 06.11.2017 15:00 Francesco Romano (LMU):Early-warning signs for critical transitions in stochastic differential equationsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In nature, we often have to deal with events that appear abruptly, characterized by a sudden change in the system we are observing. Such phenomena are known as critical transitions and can be explained as resulting from the slow variation of characteristic properties of a system. Examples are epileptic seizures, asthma, market collapses, epidemic outbursts and neuron firing. A mathematical theory of critical transitions is given in terms of bifurcations in fast-slow dynamical systems and can be used to predict such phenomena despite their abrupt nature. Based on previous works, we study some systems described by stochastic differential equations and prove that the variance of their solutions exhibits a divergent behavior as the critical transition is approached. This is done at first for linear infinite-dimensional systems, analyzing different possibilities for the spectrum of the linear operator. In particular, we relax the boundedness assumption present in previous works. Later, we prove that similar properties also hold for certain nonlinear models in finite dimensions. This suggests that an increase in the variance could be an early-warning sign for critical transitions. We use numerical simulations to show that indeed the sliding variance is a statistically relevant indicator for predicting critical transitions. We propose a possible estimator and analyze its efficiency, showing that it performs better than random guessing. Its predictive ability is also analyzed as a function of different parameters, such as the number of available data and the lead time of prediction.

TBA

### 09.11.2017 15:45 Helmut Abels (Universität Regensburg) :Diffuse Interface Models for Two-Phase Flows and Their Sharp Interface LimitsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We will discuss so-called "diffuse interface models" for the flow of two viscous incompressible Newtonian fluids in a bounded domain. Such models were introduced to describe the flow when singularities in the interface, which separates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Such kind of models became popular during the last decade for theoretical as well as numerical studies. We will give an overview of known analytic results on well-posedness and will discuss recent results on convergence of diffuse to sharp interface models.

### 09.11.2017 16:30 Timo Weidl:Semiklassische Spektralabschätzungen mit ResttermA 027 (Theresienstr. 39, 80333 München)

Ausgehend von den klassischen Li-Yau- und Berezin-Ungleichungen an die Eigenwerte des Laplace-Operators mit Dirichlet-Randbedingungen gebe ich einen Überblick über deren Verbesserung mit Resttermen in verschiedenen Situationen. Neben der Melas-Ungleichung behandeln wir deren Modifikation für Operatoren mit und ohne Magnetfeld sowie Abschätzungen mit (fast) klassischem Restterm. Abschließend erweitern wir die Ergebnisse auf den Heisenberg-sub-Laplacian und den Stark-Operator in Gebieten.

### 09.11.2017 17:15 Christian Kühn (TUM):Regularity Structures and Fractional DiffusionMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, I will first give a brief introduction to the theory of regularity structures and motivate, why it is natural to consider nonlocal operators in this context. Then, I am going to outline two applications of the theory, first to coupled SPDE-ODE systems and then to the fractional Allen-Cahn equation, which also appeared in many other contexts recently in PDE theory. The second application is also linked to graph theory, classical combinatorics, computer algebra, and mathematical physics.

### 13.11.2017 15:00 Maxime Breden (TUM):Computer-assisted proofs for dynamical systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

To understand the global behavior of a nonlinear system, the ﬁrst step is to study its invariant set. Indeed, speciﬁc solutions like steady states, periodic orbits and connections between them are building blocks that organize the global dynamics. While there are many deep, general and theoretical mathematical results about the existence of such solutions, it is often difﬁcult to apply them to a speciﬁc example. Besides, when dealing with a precise application, it is not only the existence of these solutions, but also their qualitative properties that are of interest. In that case, a powerful and widely used tool is numerical simulations, which is well adapted to the study of an explicit system and can provide insights for problems where the nonlinearities hinder the use of purely analytical techniques. However, one can do even better. Using numerical results as a starting point, and combining them with a posteriori estimates, one can then get rigorous results and prove the existence of a genuine solution close to the numerical one. In this talk, I will explain how such computer-assisted theorem can be obtained. I will then focus on some examples where these techniques can be useful, namely to study non homogeneous steady states of cross-diffusion systems, and to prove the existence of periodic solutions of the Navier-Stokes equations in a Taylor-Green ﬂow.

### 15.11.2017 16:00 Prof. Ralf Hiptmair (ETH Zürich):Shape Differentiation: New PerspectivesMI HS 3 (Boltzmannstr. 3, 85748 Garching)

The presentation examines the "derivative" of solutions of second-order boundary value problems and of output functionals based on them with respect to the shape of the domain. A rigorous approach relies on encoding shape variation by means of deformation vector fields, which will supply the direction for taking shape derivatives. These derivatives and methods to compute them numerically are key tools for studying shape sensitivity, performing gradient based shape optimization, and small-variation shape uncertainty quantification.

A unifying view of second-order elliptic boundary value problems recasts them in the language of differential forms (exterior calculus). Fittingly, the shape deformation through vector fields matches the concept of Lie derivative in exterior calculus. This paves the way for a unified treatment of shape differentiation in the framework of exterior calculus.

The obtained formulas can be employed in the so-called adjoint approach to derive shape gradients of concrete output functionals. The resulting expressions allow different reformulations. Though equivalent for exact solutions of the involved boundary value problems, they deliver vastly different accuracies in the context of finite element approximation, as confirmed by a rigorous asymptotic a priori convergence analysis for a number of important cases.

This is joint work with J.-Z. Li (SUSTC, Shenzen), A. Paganini (Mathematical Institute, University of Oxford) and S. Sargheini (Seminar for Applied Mathematics, ETH Zürich).

see http://www.ma.tum.de/Mathematik/FakultaetsKolloquium#AbstractHiptmair

### 16.11.2017 16:30 Frank den Hollander:On constrained random graphsA 027 (Theresienstr. 39, 80333 München)

http://www.mathematik.uni-muenchen.de/~mathkoll/vortraege/ws17/hollander.php

### 20.11.2017 12:30 Volkher Scholz (ETH Zürich):Analytic approaches to tensor networks for critical systems and field theoriesgroßer Hörsaal (B0.32) (Hans-Kopfermann-Str. 1, 85748 Garching)

I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically critical systems and conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that the multiscale renormalization Ansatz (MERA) does approximate conformal field theories. In particular, I will discuss the case of free fermions, both on the lattice and in the continuum, as well as Wess-Zumino-Witten models.

based on joint work with Jutho Haegeman, Glen Evenbly, Jordan Cotler (lattice) and Brian Swingle and Michael Walter (lattice & continuum)

### 20.11.2017 15:00 Christian Seis:Convergence rates for numerical approximations to linear continuity equations with rough coefficientsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider the numerical upwind scheme for approximating linear continuity equations in the low-regularity setting studied by DiPerna and Lions. Building up new stability estimates for continuity equations, we establish optimal estimates on the rate of (weak) convergence of approximating solutions towards the unique solution of the continuous problem. This is joint work with A. Schlichting (U Bonn and RWTH Aachen).

### 21.11.2017 17:00 Steffen Weißer (U des Saarlandes):Methode der Finiten Elemente auf (an-) isotropen polygonalen und polyhedralen NetzenGebäude 33, Raum 0401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In der Entwicklung numerischer Verfahren zur Approximation von Randwertaufgaben werden flexible Vernetzungen der zugrunde liegenden Gebiete immer wichtiger. Aus diesem Grund rückten sich in den letzten Jahren Verfahren in das Interesse, die auf allgemeinen polygonalen und polyhedralen Netzen anwendbar sind. Hierzu gehört auch die BEM-basierte FEM, die ihre Flexibilität durch die implizite Konstruktion der verwendeten Ansatzfunktionen erreicht. Diese werden durch lokale Randwertprobleme definiert, die im Zusammenhang mit dem globalen Problem stehen. In der computergestützten Implementierung werden die Ansatzfunktionen mit Hilfe von Randelementmethoden (BEM) realisiert.

Im bevorstehenden Vortrag wird die BEM-basierte FEM anhand des Beispieles der stationären Diffusionsgleichung mit Ansatzfunktionen höherer Ordnung eingeführt. Desweiteren werden a-priori sowie a-posteriori Fehlerabschätzungen aufgezeigt, die eine adaptive Gitterverfeinerung ermöglichen. Nachdem die bisherigen Resultate für isotrope Elemente formuliert sind, werden erste Erweiterungen für anisotrope Diskretisierungen diskutiert und Quasi-Interpolationsoperatoren eingeführt. Alle theoretischen Resultate und Überlegungen werden durch numerische Experimente bestätigt und anschaulich dargestellt, wobei insbesondere die Flexibilität der polygonalen Elemente bei adaptiver sowie anisotroper Netzverfeinerung zum Ausdruck kommt.

### 23.11.2017 15:00 Jan Bouwe van den Berg (Vrije Universiteit Amsterdam):Computer-assisted theorems in dynamics and partial differential equations: periodic solutions in the forced Navier-Stokes equationsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In nonlinear analysis we often simulate dynamics on a computer, or calculate a numerical solution to a partial differential equation. This gives very detailed, stimulating information. However, it would be even better if we can be sure that what we see on the screen genuinely represents a solution of the problem. In particular, rigorous validation of the computations would allow such objects to be used as ingredients of theorems. As an example, we focus on a rigorous numerical method to prove existence of periodic orbits in the forced Navier-Stokes equations. In the vorticity formulation we use Fourier series in both space and time. We solve for the Fourier coefficients in a Banach algebra of geometrically decaying sequences. We apply a Newton-Kantorovich type argument in a suitable neighborhood of a numerically computed candidate (the radii polynomial approach). This leads to an existence result for two-dimensional periodic orbits in the Navier-Stokes equations with so-called four vortex forcing''. This is joint work with Maxime Breden, Jean-Philippe Lessard and Lennaert van Veen.

### 23.11.2017 16:30 Sara Daneri (Universität Erlangen-Nürnberg):Patterns formation for minimizers of a local/non-local interaction functional in general dimensionRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In this talk we will consider a functional consisting of a perimeter term and a non-local term which are in competition. In the discrete setting such functional was introduced by Giuliani, Lebowitz, Lieb and Seiringer. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and one can rigorously prove that the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting had been already shown, using different techniques, by Giuliani and Seiringer. This model has many similarities with the celebrated Ohta-Kawasaki functional. In particular, for the Ohta-Kawasaki functional, the minimality of periodic stripes is conjectured. This work is in collaboration with Eris Runa.

### 27.11.2017 16:30 Dr. Timo Hirscher (Universität Stockholm) :Titel: Consensus formation in the Deffuant modelB 252 (Theresienstr. 39, 80333 München)

In 2000, Deffuant et al. introduced an interaction scheme to model opinion formation in large groups: Given a network graph and initial opinions, neighbors interact pairwise and either approach a compromise if their disagreement is below a given threshold or ignore each other if not. Concerning the asymptotics of the model, one central question is whether the whole group achieves a general consensus in the long run or splits into irreconcilable parts.

We studied this model featuring univariate opinions on integer lattices and infinite percolation clusters as underlying interaction networks. The generalization of the original model to multivariate and measure-valued opinions was analyzed on the simple network given by the doubly-infinite path Z.