Idisplays

In the context of fairly simple elliptic partial differential equations, overdetermined boundary conditions arise in the search of optimal shapes in a broad range of problems, e.g., in fluid mechanics, the theory of elasticity, electrostatics and integral geometry. Due to their relevance, the resulting overdetermined boundary value problems are addressed in prominent conjectures. The Berestycki-Caffarelli-Nirenberg conjecture from 1997, disproved by Sicbaldi in 2010, has lead to various recent results on the existence and classification of extremal unbounded domains. These unbounded optimal shapes can be regarded as analogues of constant mean curvature surfaces governed by nonlocal effects. Schiffer?s conjecture, and the related Pompeiu problem in integral geometry from 1929, are still open. In my talk, I will discuss a choice of classical and recent results on overdetermined boundary value problems, including joint work with M.M. Fall and I.A. Minlend.

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions. This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

Further information

Zahra Mohammadi Khangheshlaghi: Model Theory and The Free Factor Complex
Hanna Stange: Hyperuniformity and Rigidity of Perturbed Lattices
Alex Tullini: From elliptic to hyperbolic PDEs: the things I did for the love of black holes

We study the (discrete) Phi^4 model on Z^d with algebraically decaying long-range interactions. This model is the natural discrete analogue of the fractional continuous Phi^4 model. We first describe some general properties of the phase transition that the model undergoes. Then, we study the nature of the critical scaling limits of the model, proving in particular the (marginal) triviality of these limits in dimension 3 with well-chosen interactions.

Jacques Tits generalised the notion of a "polarity" in order to describe all embeddable polar spaces. We further extend it to the notion of "generalised duality". We use it to classify and describe all geometric hyperplanes of a Segre geometry, which is the direct product of two arbitrary projective spaces. This result, in turn, can be applied to Segre varieties over arbitrary fields and we obtain an explicit list of all geometric hyperplanes that are not induced by a projective hyperplane. Among them are so-called black hyperplanes, which are embedded long root geometries of type A, and we will mention some special features about those.