Veranstaltungen

In this talk I will present some recent results concerning non-radial implosions for compressible Euler and Navier-Stokes equation and non radial blow up for certain defocusing supercritical nonlinear Schrodinger equations. This work is a non radial generalization of the breakthrough results of Merle-Raphael-Rodnianski-Szeftel. The work presented is in collaboration with Gonzalo Cao-Laboratories, Javi Gomez-Serrano and Jia Shi.

A mathematically rigorous description of fluid motion plays an important role, for example in understanding weather patterns or ocean dynamics. In this talk, we will derive the fundamentals equations of motions for an inviscid incompressible fluid -- the Euler equations. We will present known results as well as related open problems. Finally, we will discuss stability for 2D stratified flows relevant in geophysics.

Two non-isomorphic ergodic measure preserving flows can become isomorphic if one of the systems undergoes an appropriate time change. In this case we will say that these flows are Kakutani equivalent to each other. We say that an ergodic flow is loosely Kronecker if it is Kakutani equivalent to the straight line flow on (say) a two torus in an irrational direction (the exact direction is immaterial as these are all equivalent to each other). Landmark work of Ratner from the late 70s (that paved the way to her even more famous results on orbit closures and equidistribution of unipotent flows) establishes that 1) the horocycle flow on any finite area surface of constant negative curvature is loosely Kronecker. 2) the product of two such flows is not loosely Kronecker. It remained an open problem whether e.g. products of two horocycle flows are Kakutani equivalent to each other. We show unipotent flows are very rigid under time changes, and indeed unless the flows are loosely Kronecker, two unipotent flows are Kakutani equivalent if and only if they are isomorphic as measure preserving flows. This is a joint work with Elon Lindenstrauss

There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.

In this talk, we will focus on solving time-harmonic, acoustic, elastic and polarized electromagnetic waves scattered by multiple finite-length open arcs in unbounded two-dimensional domain. We will first recast the corresponding boundary value problems with Dirichlet or Neumann boundary conditions, as weakly- and hyper-singular boundary integral equations (BIEs), respectively. Then, we will introduce a family of fast spectral Galerkin methods for solving the associated BIEs. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We will show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arc geometries. Numerical examples will demonstrate the accuracy and robustness of the proposed methods with respect to number of arcs and wavenumber. Moreover, we will show that for general weakly- and hyper-singular boundary integral equations their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape holomorphy of domain-to-solution maps associated to BIEs appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications. Also, they pose new questions you may have the answer to!

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity θ ∈ (0, 1/2]. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii r and rs as r → 0 (s > 1 fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius r decays like | log r|−1+θ+o(1) as r → 0. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of logα-capacity.

Three well-studied (classes of) objects in low-dimensional topology are knots (in various settings), mapping class groups of manifolds, and motion groups. We will see how these can be related to each other, and illustrate these connections using the example of braids and links. We will then introduce the Goeritz group, which may be thought of as a higher-dimensional analogue of the (spherical) braid group, and conclude with open problems around it.

The skein module of a 3-manifold is a rich algebraic object whose elements are knots and links; it has many fascinating connections to representation theory, mathematical physics and the Jones polynomial. In this talk I will give a brief introduction to the topic, including discussing the sort of interesting problems that come up with these objects, and then I will focus on a recent joint work with R. Detcherry about the relationship between torsion in the skein module and interesting surfaces in the manifold.

Pascals “Traité du triangle arithmétique” wurde zusammen mit “quelques autres petits traitez sur la mesme matiére” veröffentlicht: eine Perlenkette an verschiedenartigen Abhandlungen mit einem kombinatorischen Leitfaden. Diesen rhapsodischen Stil wollen wir weiterführen. Unser Startpunkt wird eine Eigenschaft des arithmetischen (“Pascal’schen”) Dreieckes sein, die womöglich zu den weniger bekannten zählt. Ausgehend davon werden wir eine kleine mathematische Reise unternehmen, die Themen wie Graphenfärbungen, Polynome mit reellen Nullstellen und Vektorgeometrie verbinden wird. Am Ende werden wir das Resultat illustrieren können, für das June Huh 2022 die Fields-Medaille erhalten hat.

We propose an efficient, reliable, and interpretable global solution method, the Deep learning-based algorithm for Heterogeneous Agent Models (DeepHAM), for solving high dimensional heterogeneous agent models with aggregate shocks. The state distribution is approximately represented by a set of optimal generalized moments. Deep neural networks are used to approximate the value and policy functions, and the objective is optimized over directly simulated paths. In addition to being an accurate global solver, this method has three additional features. First, it is computationally efficient in solving complex heterogeneous agent models, and it does not suffer from the curse of dimensionality. Second, it provides a general and interpretable representation of the distribution over individual states, which is crucial in addressing the classical question of whether and how heterogeneity matters in macroeconomics. Third, it solves the constrained efficiency problem as easily as it solves the competitive equilibrium, which opens up new possibilities for normative studies. As a new application, we study constrained efficiency in heterogeneous agent models with aggregate shocks. We find that in the presence of aggregate risk, a utilitarian planner would raise aggregate capital for redistribution less than in absence of it because poor households do more precautionary savings and thus rely less on labor income. Joint work with Jiequn Han and Weinan E