Termine

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.

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.

Pro-etale cohomology of rigid-analytic spaces with Q_p-coefficients has some surprising features: it is not A^1-invariant and no general finiteness theorems over Q_p are true. It has been observed in recent years that these particularities can be explained by viewing the pro-etale cohomology as (quasi-)coherent cohomology on the Fargues-Fontaine curve. I want to explain joint work in progress with Arthur-Cesar Le Bras and Lucas Mann, which aims to fully implement this idea by developing a six functor formalism with values in solid quasi-coherent sheaves on relative Fargues-Fontaine curves.

The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. On Kerr black holes, the analysis of Maxwell's equations and the equations of linearized gravity, can be simplified by introducing the Teukolsky equation, which offers the advantage of being scalar in nature. After explaining this reduction, I will present a result providing the large time leading-order term for initially localized and regular solutions of the Teukolsky equation, valid for the full subextremal range of black hole parameters and for all spins. I will explain how such a development follows naturally from the precise analysis of the resolvent operator on the real axis. Recent advances in microlocal analysis are used to establish the existence and mapping properties of the resolvent.

The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford?s theorem, [11], [8]). Kinetic theory may also be fruitfully used to model large systems of particles that interact inelastically (sand, snow, interstellar dust, see [3], [10], [4]). Such materials are known as granular media. The theory enables for instance to explain the onset of inhomogeneities, as well as to quantify the decay of the temperature (Haff?s law, [9], [3]). In this case, the derivation of the inelastic Boltzmann equation is still open, mainly due to the complicated dynamics of the particles. In particular, it is still unknown if the dynamics of such particle systems is well-posed. One major difficulty comes from the phenomenon of inelastic collapse. A system of particles is said to experience an inelastic collapse when infinitely many collisions take place in finite time. It is known that inelastic collapse may take place for systems of only three particles [12]. We studied systems of three particles, in dimension d ? 2. Assuming that the restitution coefficient r is constant, we obtained general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff [13], and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision. Another important question is the following: since inelastic collapse can take place, is it possible to continue the dynamics of the particles anyway? We report also partial results in this direction. Considering on the other hand another law of collision, prescribing that a fixed quantity of kinetic energy is lost during each collision, we obtained results on systems of an arbitrary number of particles interacting according to this law, that look a priori contradictory. Namely, we proved that the flow of such a system of particles conserves the measure in the phase space, whereas the kinetic energy is not conserved. From these results, we deduce an Alexander?s theorem [1] for such systems of particles: for almost every initial datum, the dynamics of such systems is globally well-posed. To the best of our knowledge, this is the first result of global well-posedness concerning the dynamics of systems of inelastic particles. The results are taken from [5], [7], [6], obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).