Termine

This colloquium will explore some fundamental issues in the mathematics of plasmas, focusing on the stability and instability of solutions to Vlasov-type equations, which are crucial for describing the behavior of charged particles in a plasma. A general introduction to kinetic theory is given, making the subject accessible to a wide audience of mathematicians. Key mathematical concepts such as well-posedness, stability, and the behavior of solutions in singular limits are discussed. In addition, a new class of Wasserstein-type distances is introduced, offering new perspectives on the stability of kinetic equations.

Given an action of a group G on a relational Fraïssé structure M, we call this action sharply k-homogeneous if, for each isomorphism f : A -> B of substructures of M of size k, there is exactly one element of G whose action extends f. This generalises the well-known notion of a sharply k-transitive action on a set, and was previously investigated by Cameron, Macpherson and Cherlin. I will discuss recent results with J. de la Nuez González which show that a wide variety of Fraïssé structures admit sharply k-homogeneous actions for k ? 3 by finitely generated virtually free groups. Our results also specialise to the case of sets, giving the first examples of finitely presented non-split infinite groups with sharply 2-transitive/sharply 3-transitive actions.

For a topological group G, let Aut(G) be the group of all automorphisms, and let w(G) be the number of orbits of Aut(G) on G. I will report about results characterizing classes of groups G with suitable (topological or algebraic) assumptions on G, combined with bounds on w(G). In particular, there are strong results in the following cases: * G is locally compact and connected, with w(G) less than the cardinality of the continuum * G is compact, and w(G) is finite * G is finite, and w(G) < 5

Markus Stroppel will give a lecture series on advanced topics in locally compact groups. The lectures take place on Monday at 14:15 in room SR 1D They will start on Monday, November 17. The planned topics are: * the scale function and tidy subgroups * automorphism groups of trees

Abstract: The bottom homotopy group of a flat commutative algebra over the sphere spectrum inherits a delta-ring structure from the Tate-valued Frobenius of Nikolaus-Scholze, which plays a big role in the relationship between trace methods and prismatic cohomology. It is an interesting question how much of the E_infty ring structure is remembered by this delta-ring structure. In this talk, we present a modest step towards understanding this, by proving that there is a one-to-one correspondence between H_infinity ring structures on flat S-modules and delta-ring structures on their pi_0.

Abstract: Tian's classical theorem shows that, for a polarized complex projective manifold, the Bergman kernel grows asymptotically like a monic polynomial of degree equal to the dimension. I will present a weighted extension of this result that includes, for instance, equivariant and divisor-adapted Bergman kernels. In contrast to the classical case, the leading coefficient now reflects the speed of a geodesic ray in Mabuchi geometry associated with the weight. In the equivariant case, this speed coincides with the Hamiltonian of the induced circle action, while in the divisor case, it arises via a resolution of the Monge-Ampère equation on the deformation to the normal cone. Finally, I will discuss how this equidistribution result connects to the theory of canonical metrics, and in particular, its implications for the Wess-Zumino-Witten equation.

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon\in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as ε $\to 0^+$, and show that the corresponding solutions $\omega_ε(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. Based on joint work with Alberto Enciso and Antonio Fernandez.

Xmas PDE Colloquium - coffee break after (14:45 - 15:15, SRZ Lounge)