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Following work of Evert, Helton, Klep and McCullough on free linear matrix inequality domains, we ask when a matrix convex set is the closed convex hull of its (absolute) extreme points. This is a finite-dimensional version of Arveson‘s non-commutative Krein-Milman theorem, which may generally fail completely since some matrix convex sets have no (absolute) extreme points. In this talk we will explain why the Arveson-Krein-Milman property for a given matrix convex set is difficult to determine. More precisely, we show that this property for certain commuting tensor products of matrix convex sets is equivalent to a weak version of Tsirelson‘s problem from quantum information. This weak variant of Tsirelson‘s problem was shown, by a combination of results of Kirchberg, Junge et. al., Fritz and Ozawa, to be equivalent to Connes‘ embedding conjecture; considered to be one of the most important open problems in operator algebras. We do more than just provide another equivalent formulation of Connes‘ embedding conjecture. Our approach provides new matrix-geometric variants of weak Tsirelson type problems for pairs of convex polytopes, which may be easier to rule out than the original weak Tsirelson problem.

Based on joint work with Roy Araiza and Thomas Sinclair

For a quasi-split group $G$ over a local field $F$, with Borel subgroup $B=TU$ and Weyl group $W$, there is a natural geometric action of $G\times T$ on $L^2(X),$ where $X=G/U$ is the basic affine space of $G$. For split groups, Gelfand and Graev have extended this action to an action of $G\times (T\rtimes W)$ by generalized Fourier transforms $\Phi_w$. We shall extend this result for quasi-split groups, using a new interpretation of Fourier transforms for quasi-split groups of rank one.

This is joint work with David Kazhdan.

Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a Riemann surface.

In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows.

Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold.