Abstract: The profinite completion of a free profinite group on an infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group as well. I am going to present some partial results regarding this question.
Let Γ be a discrete group. A group Γ is called unitarisable if for any Hilbert space H and
any uniformly bounded representation π : Γ → B(H) of Γ on H there exists a bounded operator
S : H → H such that S^{−1}π(g)S is a unitary representation for any g ∈ Γ. It is well known that
amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups.
Dixmier: Are all unitarisable groups amenable?
One of the approaches to study unitarisability is related to the space of the Littlewood functions
Title - Measure rigidity of Anosov flows via the factorization method.
Abstract: 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.
Abstract: In the 70s, physicists proposed several models fordisordered magnetic alloys called spin glass models. Mathematically, thespherical models are random functions on the sphere in high-dimensions, andmany of the questions physicists are interested in can be phrased aspurely mathematical questions about geometric properties, extreme values,critical points, and Gibbs measures of random functions and the interplaybetween them.
Abstract: Host proved a strengthening of Rudolph and Johnson's measure rigidity theorem: if a probability measure is invariant, ergodic and has positive entropy for the map x2 mod 1, then a.e. point equidisitrbutes under x3 mod 1. Host also proved a version for toral endomorphisms, but its hypotheses are in some ways too strong, e.g. it requires one of the maps to be expanding, so it does not apply to pairs of automorphisms. In this talk I will present an extension of Host's result (almost) to its natural generality.
