Hausdorff Kolloquium 2022

Dates: April 20 and July 6, 2022

Organizers: Ursula Hamenstädt and Juan Velázquez

 

Venue: Lipschitzsaal, Mathezentrum, Endenicher Allee 60, 53115 Bonn

The first talk on April 20th and the first talk on July 6th will be held online (see Zoom data below). The second talk on April 20th and the second talk on July 6th will be held on-site.

Abstracts

Anne-Laure Dalibard (Sorbonne Université): Around the stationary Prandtl equation

Viscous incompressible fluids are described by the Navier-Stokes system. For fluids with small viscosity in the vicinity of a solid obstacle, Ludwig Prandtl predicted that a boundary layer (i.e. a zone of small width in which the velocity of the fluid changes abruptly) is created close to the boundary of the obstacle. He also derived a system of equations for the fluid within the boundary layer. This system has been a topic of intense research in the past decades. In this talk, we will focus on the stationary setting, and we will make a review of the existing results and of some open problems. I will also give some insight of the proofs.

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Alan Reid (Rice University): Profinite rigidity, direct products and finite presentability

A finitely generated residually finite group G is called profinitely rigid, if for any other finitely generated residually finite group H, whenever the profinite completions of H and G are isomorphic, then H is isomorphic to G. In this talk we will review what is known about this in the context of groups arising in low-dimensional geometry and topology. We will review recent examples of lattices in PSL(2,R) and PSL(2,C) that are profinitely rigid, and discuss how the techniques can be used to produce examples of finitely presented groups that are profinitely rigid amongst finitely presented groups but not amongst finitely generated one.

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Angkana Rüland (Universität Heidelberg): On the (fractional) Calderon problem

Inverse problems are ubiquitous in nature, engineering and our daily lives. A prototypical nonlinear inverse problem is the Calderon problem which arises in electrical impedance tomography and in which one seeks to recover an unknown conductivity within a conducting body from voltage-to-current measurements at its surface. Major open questions concern the uniqueness, stability and reconstruction of the conductivity for irregular or anisotropic conductivities as well as settings in which only partial data are given. In this talk I will introduce this inverse problem and illustrate how nonlocality and associated uncertainty principles allow to solve some of these questions in the setting of the fractional Calderon problem, a nonlocal analogue of this problem.

Barak Weiss (Tel Aviv University): Geometric and arithmetic aspects of approximation vectors

I will describe several natural questions that arise in connection with the sequence of best approximation vectors. This sequence, which I will introduce in the talk, is a higher dimensional generalization of continued fraction convergents (which I will also discuss briefly). The questions involve the statistical behavior of certain observables, and the means to understand them involve a mix of number theory, ergodic theory, and elementary geometry. The talk will be based on joint work with Uri Shapira.