Abstract: A permutation representation of a group G is called highly transitive if it is transitive on k-tuples of points for every k. Until just a few years ago groups admitting such permutation representations were thought of as rare. I will focus on three rather recent papers: G-Garion, Hall-Osin, Gelander-G-Meiri (in preparation) showing that such groups are in fact very common.
(joint work with Françoise Dal'Bo and Andrea Sambusetti)
Given a finitely generated group G acting properly on a metric space X,
the exponential growth rate of G with respect to X measures "how big"
the orbits of G are. If H is a subgroup of G, its exponential growth
rate is bounded above by the one of G. In this work we are interested in
the following question: what can we say if H and G have the same
exponential growth rate? This problem has both a combinatorial and a
geometric origin. For the combinatorial part, Grigorchuck and Cohen
The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K).
In this talk we present a proof of the Kodaira's theorem that gives a sufficient condition on the existence of an embedding of a Kahler manifold into CPn. This proof is based on the Kodaira Vanishing theorem, using a sheaf-cohomological translation of the embedding conditions.
לאירוע הזה יש שיחת וידאו.