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# Summer term 2021

## July 15, 2021

16:15 -17:45 (A&G)

Prof. Raffaele Vitolo (Univesitá del Salento, Italy)

### Hamiltonian operators, quasilinear PDEs and projective geometry

Abstract: The Hamiltonian formulation of quasilinear first-order partial differential equations, or hydrodynamic-type systems, has well-known invariance properties with respect to differential-geometric transformations of coordinates. In this talk we will discuss recent results that show the presence of projective-geometric invariance in both Hamiltonian operators and quasilinear first-order PDEs.

(Contact: V. Matveev / M. Dafinger)

## July 1, 2021

16:15 -17:45 (DS&MP)

Prof. Mostafa Sabri (Cairo University)

### Spectral analysis of large quantum graphs

Abstract: Quantum graphs are continuum analogs of discrete (combinatorial) graphs in which the adjacency matrix is replaced by a differential operator on the edges. They arise naturally in chemistry, physics and engineering when one considers propagation of waves through a quasi-one-dimensional system, and they are also interesting for purely mathematical reasons.

This talk is intended for a wide audience; I will first introduce quantum graphs then discuss four problems : how to define an appropriate notion of convergence for sequences of quantum graphs ? Does this notion guarantee the convergence of empirical spectral measures ? How about the nature of the spectrum at the limit infinite quantum graph ? Can we show that in some cases it exhibits absolutely continuous spectrum ? And if yes, what if we consider a sequence of finite quantum graphs converging to such model, are the eigenvectors delocalized in the sense that they are essentially uniformly distributed over the graph ?

This is based on several joint works with Nalini Anantharaman, Maxime Ingremeau and Brian Winn.

(Contact: D. Lenz)

## June 24, 2021

16:15 -17:45 (DS&MP)

Dr. Reza Mohammadpour Bejargafsheh (University of Bordeaux)

### Ergodic optimization and multifractal formalism of Lyapunov exponents

Abstract: You can find the pdf file here.

In this talk, we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type, and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show both the continuity of the entropy at the boundary of the Lyapunov spectrum for such cocycles and the continuity of the minimal Lyapunov exponent under the assumption that linear cocycles satisfy a cone condition.

We consider a subadditive potential $\Phi$. We obtain that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure, and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures.

(Contact: D. Lenz)

## June 17, 2021

16:15 -17:45 (A&G)

Prof. Gabriele Mondello (Sapienza – Università di Roma)

### On spherical surfaces of genus 1 with 1 conical point

Abstract: A spherical surface with conical points is real 2-dimensional manifold that can be obtained from a disjoint union of convex spherical triangles by isometric identification of pairs of edges. Thus, to every spherical surface we can associate an underlying Riemann surface with marked points. Once the angles are fixed (and the Gauss-Bonnet constraint permits), the analogous procedure with hyperbolic or flat surfaces produces a bijection between constant curvature metric structures with conical singularities and conformal structures with marked points, which generalizes the uniformization theorem. In the spherical case the situation looks quite different.

In this talk I will first review some known results about the topology of the moduli space of spherical surfaces of genus g with n conical points. Then I will describe a synthetic approach to the case of genus 1 with 1 conical point in more detail, which is joint work with Eremenko-Panov.

(Contact: V. Matveev / M. Dafinger)

## June 10, 2021

16:15 -17:45 (A&G)

Prof. Ian Anderson (Utah State University)

### Spacetimes with Symmetry

Abstract: The mathematical physicist A. Z. Petrov, who is best known for his classification of spacetimes according to the algebraic character of the Weyl tensor, also gave a remarkable classification of local spacetimes which admit a non-trivial Lie algebra of Killing vector fields. In this talk I will focus on two open problems arising from Petrov’s classification of spacetimes with symmetry.

The first open problem deals with spacetimes whose isometry group is simply transitive. Using the Newman-Penrose formalism, I will describe a complete algebraic enumeration of all such spacetimes and a detailed solution to the local equivalence problem. The effectiveness of this classification is demonstrated by solving some long-standing problems in the field of exact solutions of the Einstein equation.

The second problem focuses on the classification of spacetimes with intransitive isometry groups for which there is no slice. I’ll describe a new method for the local classification of these spacetimes.

(Contact: V. Matveev / M. Dafinger)

## June 3, 2021

16:15 -17:45 (A&G)

Prof. Vsevolod Shevchishin (University of Warmia and Mazury in Olsztyn)

### Polynomially superintegrable metrics on surfaces admitting a linear integral.

Abstract:We give a complete local classification of superintegrable metrics on surfaces admitting two polynomial integrals one of which is linear.
We also describe the Poisson algebra of polynomial invariants of such metrics presenting a natural set of generators and polynomial relations between them and giving expressions of Poisson brackets of those generators.
We also give explicit formulas in some interesting cases.

(Contact: V. Matveev / M. Dafinger)

## May 27, 2021

16:15 -17:45  (DS&MP)

Max Kämper (TU Dortmund)

### Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory

Abstract: Random Schrödinger operators are a model for metals with random impurities and this talk will present them for the special case of the Anderson operator on a lattice. We will introduce the integrated density of states and its uniform approximation by eigenvalue counting functions and show how results from empirical process theory can be used to improve quantitative results for this approximation.
This talk is based on joint work with Christoph Schumacher, Fabian Schwarzenberger and Ivan Veselic.

(Contact: D. Lenz)

## May 20, 2021

16:15 -17:45 (A&G)

Prof. Gérard Besson (Institut Fourier, Universität Grenoble Alpes)

### Finiteness Theorems for Gromov-Hyperbolic Groups

Abstract: You can find the pdf file here.

This is a joint work with G. Courtois, S. Gallot and A. Sambusetti.
We shall prove that, given two positive numbers $\delta$ and $H$, there are finitely non cyclic torsion-free $\delta$-hyperbolic marked group $(\Gamma , \Sigma)$ satisfying ${\rm Ent} (\Gamma , \Sigma) \le H$, up to isometry (of marked groups). Here a marked group is a group $\Gamma$ together with a symmetric generating set $\Sigma$ and ${\rm Ent}$ is the entropy of the marked group. These notions will be defined precisely.

(Contact: V. Matveev / M. Dafinger)

## May 12, 2021

12:15-13:45 (A&G)

Nicolas Boumal (Ecole Polytechnique Fédérale de Lausanne (EPFL))

### Riemannian geometry for numerical optimization: the tools we use and some geometry questions that arise

(Contact: V. Matveev / M. Dafinger)

## May 6, 2021

16:15 -17:45 (A&G)

Davide Parise (University of Cambridge)

### Convergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n - 2)-area functional

Abstract: We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern.

The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. In the first part of the talk we will give a general overview of the codimension one theory. We will then move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied as a basic model problem in gauge theory. We will explain to what extend the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interprete the former as a relaxation/regularization of the latter. Time permitting we will mention some elements of the proof, with special emphasis on the gradient flow of the Yang-Mill-Higgs energies.

(Contact: V. Matveev / M. Dafinger)

## April 29, 2021

16:15 -17:45 (A&G)

Dr. Dong Cheng & Prof. Dimitri Burago (Pennsylvania State University)

### Open problems in geometry

(Contact: V. Matveev / M. Dafinger)

## April 22, 2021

16:15 -17:45 (A&G)

Prof. Alexey Glutsyuk (CNRS, ENS de Lyon; HSE University (Moscow))

### On polynomially integrable billiards on surfaces of constant curvature

Abstract: You can find the pdf file here.

(Contact: V. Matveev / M. Dafinger)

## April 15, 2021

16:15 -17:45 (A&G)

Dr. Louis Merlin  (RWTH Aachen University)

### On the relations between the universal Teichmuller space and Anti de Sitter geometry

Abstract: Anti de Sitter (AdS) space is the Lorentzian cousin of the hyperbolic 3-space: it is a symmetric space with constant curvature -1. In this talk, we will consider surface group representations in the isometry group of AdS space, called quasi-Fuchsian representations. There is 2 classical objects associated to those representations and one of the goal is to understand their interplay: the limit set which is a quasi-circle in the boundary at infinity of AdS space and a convex set inside AdS which is preserved by the group action and bounded by two pleated surfaces. I will conclude the talk by a report on a work in common with Jean-marc Schlenker where we extend the "Teichmüller" situation to the "universal Teichmüller".

(Contact: V. Matveev / M. Dafinger)

## April 8, 2021

16:15 -17:45 (A&G)

Marcos Cossarini  (Ecole Polytechnique Fédérale de Lausanne (EPFL))

### Discrete surfaces with length and area and minimal fillings of the circle

Abstract: We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of crossings with the walls, and the area of the surface is the number of crossings between the walls themselves. We show how to approximate any Riemannian (or self-reverse Finsler) metric by a wallsystem.

This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere has minimum area among orientable Riemannian surfaces that fill isometrically a closed curve of given length. (A surface fills its boundary curve isometrically if the distance between each pair of boundary points measured along the surface is not less than the distance measured along the curve.) We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n-1)/2 squares. This conjecture is equivalent to the continuous FAC for surfaces with self-reverse Finsler metric.

(Contact: V. Matveev / M. Dafinger)

## April 1, 2021

16:15 -17:45 (A&G)

Prof.  Andrey Mironov (Sobolev Institute of Mathematics, Novosibirsk, Russia)

### On integrable magnetic geodesic flows on 2-torus

Abstract: We will discuss the problem of integrability of geodesic and magnetic geodesic flows on 2-torus. The talk is based on joint results with Misha Bialy (Tel-Aviv) and Sergey Agapov (Novosibirsk).

(Contact: V. Matveev / M. Dafinger)