Alternatively have a look at the program.

## Diffeomorphism groups, moduli spaces, and Ricci flow. Part 1

The lecture will explain some new applications of Ricci flow to long-standing conjectures concerning the topology of diffeomorphism groups and moduli spaces of Riemannian metrics.

## Log-concavity of volume

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.

## Program discussion

## On systolic growth of Lie groups

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.

## Harmonic surfaces in 3-manifolds and the simple loop theorem

Denote by ${\eufm M}(\Sigma)$ the space of hyperbolic metrics on a closed, orientable surface $\Sigma$ and by ${\eufm M}(M)$ the space of negatively curved Riemannian metrics on a closed, orientable 3-manifold $M$. We show that the set of metrics for which the corresponding harmonic map is in Whitney's general position is an open, dense, and connected subset of ${\eufm M}(\Sigma)\times {\eufm M}(M)$. The main application of this result is the proof of the Simple Loop Theorem for hyperbolic 3-manifolds. Consequences regarding minimal surfaces will be mentioned.

## Diffeomorphism groups, moduli spaces, and Ricci flow. Part 2

The lecture will explain some new applications of Ricci flow to long-standing conjectures concerning the topology of diffeomorphism groups and moduli spaces of Riemannian metrics.

## Existence of infinitely many minimal hypersurfaces in closed manifolds

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.

## Program discussion

## Complex hyperbolic surfaces with cusps

## Compactifications of Hitchin and maximal character varieties

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