Termine

wissenschaftliches Kolloquium am IDMI

I'll give an overview of stable rank one and the C*-algebras known to have it, then describe a new approach to studying stable rank one in possibly nonnuclear crossed products. As an application, we show that stable rank one is generic for two natural classes of minimal actions of free groups on the Cantor set. Time permitting, I will discuss some examples to motivate parts of the proof. This is joint work with Shirly Geffen and David Kerr.

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

The family of Zolotarev distances between two probabilities naturally extends the Wasserstein-1 metric to higher orders: one bounds the Lipschitz constant of the relevant derivatives of the potential. So far, however, the optimal transport perspective has been available only for the first order. In my talk I will demonstrate how a PDE motivation revolving around optimal elastic structures has led us to a new (OT) framework for the second-order Zolotarev distance. The new duality theory paves a way to efficient computational method, including a variant of the famous Sinkhorn algorithm. The talk is partially based on a joint work with Guy Bouchitté (Université de Toulon).

A natural generalization of the Lebesgue sequence space to the non-commutative setting of n by n-matrices is to take the p-norm of the singular values, the Schatten p-norm. Thus, in case of p equal to one we obtain the nuclear/trace norm, in case of p equal to two the Hilbert-Schmidt/Frobenius norm and in case of p equal to infinity the spectral norm. In this talk, we will be interested in the volume of Schatten p-balls, that is, unit balls with respect to Schatten p-norms, as their dimension tends to infinity. In contrast to the commutative analog, the finite-dimensional Lebesgue sequence space for which the volume of the unit ball is known since Dirichlet, exact results are only available for p equal to two and to infinity. Asymptotics for general p have been established starting in the 80's with two very recent contributions in the last month. We give insight on underlying connections to the theory of random matrices, logarithmic potentials and statistical mechanics. In particular, we highlight a large deviation principle due to Leblé and Serfaty from which we obtain finer asymptotics for the volume of Schatten $p$-balls in the self-adjoint case. Perhaps interestingly, both the differential entropy and its noncommutative analog - the free entropy - of the corresponding equilibrium measure appear.

Abstract: For a direct sum of line bundles, the symmetric power with respect to such a tableau can be expressed using the corresponding Schur polynomial. This is the starting point for passing to K-theory.

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.

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.