Termine

Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai?s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry?s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

The Bernstein decomposition makes precise how the category of smooth complex representations of a p-adic reductive group G is built from supercuspidals. The center of the category factors accordingly and is largely well-understood. If one replaces the complex numbers by a coefficient field of characteristic p, the analogous category has strikingly different features. For one, the (Bernstein) center is rather small and often determined by the center of G. This was shown by Ardakov and Schneider for a general locally pro-p group G (under a mild hypothesis). In this talk, I will discuss the corresponding question in the derived setting. I will present various results related to the graded center of the derived category of smooth mod p representations. The talk is based on joint work (in progress) with Peter Schneider.

Bounded cohomology is a functional-analytic variant of ordinary cohomology with applications in the study of manifold geometry, rigidity theory, ergodic theory, and many other areas. The aim of this talk is to discuss a result concerning the vanishing of the bounded cohomology of the transformation groups of Euclidean spaces. To this end, we will first recall the definition of (bounded) cohomology of groups, along with its main properties and key results. We will then briefly review the notion of amenability and explore its connection with bounded cohomology. We will then focus on the aforementioned result. We will begin with the group of homeomorphisms of R^n with compact support, as treated by Matsumoto and Morita in 1985, and then turn on the full groups of all the homeomorphisms and diffeomorphisms of Euclidean spaces. The latter result is recent and is due to Fournier-Facio, Monod, and Nariman (2024). Quite surprisingly, in both cases the bounded cohomology vanishes, although the techniques employed are substantially different

In forthcoming joint work with Christoph Kehle, we present a PDE-based proof that for any Einstein--Maxwell--matter system satisfying the local mass-charge inequality, corresponding to $\mathfrak{m}^2 \geq \mathfrak{e}^2$ for charged massive dust, any solution arising in gravitational collapse from spherically symmetric data on $\mathbb R^3$ cannot settle down (in a suitable sense) to an extremal Reissner--Nordström black hole. This generalizes a recent result of Reall (2024), who proved, using spinor methods, that extremal Reissner--Nordström cannot form in finite advanced time if the matter satisfies the mass-charge inequality. Moreover, we prove the sharpness of the mass-charge inequality: for any $\mathfrak{m}^2<\mathfrak{e}^2$, we construct smooth, one-ended, asymptotically flat Cauchy data for the Einstein--Maxwell massive charged dust model whose maximal future development contains a black hole with an event horizon that becomes exactly extremal Reissner--Nordström at finite advanced times.

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given at first order by classical dynamics, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to Klein-Gordon equations and how to derive relativistic mechanics (instead of classical mechanics) at their semi-classical limit.

Rather surprisingly, the technique of convex integration has revealed that for some initial data, many models studied in the field of mathematical fluid mechanics allow for a multitude of solutions. This talk is concerned with the question how large the set of such wild initial data is in the context of the isentropic Euler equations. We show that wild initial data form a dense set. In contrast to existing results in the literature, in this talk "wild initial data" are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds.

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space's boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group's quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths. In this talk, I will give an insight into classical Patterson-Sullivan theory, and show how studying bi-infinite random walks on a discrete hyperbolic group $G$ leads to an analogue of the Patterson-Sullivan measure on $\partial^2G$. This talk is based on joint work with Mahan Mj and Chiranjib Mukherjee.