Abstracts for Arbeitstagung 2019 on Geometry

In this talk we present a proof of the log-concavity property of total masses of positive currents on a given compact Kähler manifold, that was conjectured by Boucksom, Eyssidieux, Guedj and Zeriahi. The proof relies on the resolution of complex Monge-Ampère equations with prescribed singularities. As corollary we give an alternative proof of the Brunn-Minkowsky inequality for convex bodies. This is based on a joint work with Tamas Darvas and Chinh Lu.

Introduced by Gromov in the nineties, the systolic growth of a
finitely generated group maps $n$ to the smallest index of a finite
index subgroup meeting the $n$-ball only in the identity singleton.
This function is one measure of residual finiteness. It extends to
compactly generated locally compact groups, replacing "finite index"
with "cocompact lattice" in the definition.

It grows as least as fast as the word growth, and with Bou-Rabee we
showed that the growth is exponential for linear groups of exponential
growth.

In the early 80's, Yau conjectured that in any closed $3$-manifold there should be infinitely many closed minimal surfaces. I will survey previous results related to the question and present a proof of the conjecture. It builds on the min-max theory of F. Almgren and J. Pitts, which has recently been further developed by F. C. Marques and A. Neves.