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.
In this talk we study a natural generalization of the classical \eps-net problem (Haussler-Welzl 1987), which we call 'the \eps-t-net problem': Given a hypergraph on n vertices and parameters t and \eps , find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least \eps n contains a set in S. When t=1, this corresponds to the \eps-net problem.
Title: A randomized construction of high girth regular graphs
Abstract: We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k > 2$ and $0 < c < 1$ be fixed. Let $n$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices. As long as the smallest degree $\delta (G)<k$, choose, uniformly at random, two vertices $u,v \in V(G)$ of degree $\delta(G)$ whose distance is at least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge $uv$ to $E(G)$.
