# Geometria em Lisboa Seminar

## Planned sessions

### Probabilistic aspects of toric Kahler geometry

Let $(M, \omega, L)$ be a polarized toric Kahler manifold with polytope $P$. Associated to this data is a family $\mu_k^x$ of probability measures on $P$ parametrized by $x \in P.$ They generalize the multi-nomial measures on the simplex, where $M = \mathbb{CP}^n$ and $\omega$ is the Fubini-Study measure. As is well-known, these measures satisfy a law of large numbers, a central limit theorem, a large deviations principle and entropy asymptotics. The measure of maximal entropy in this family corresponds to the center of mass $x$ of $P$. All of these results generalize to any toric Kahler manifold, except the center of mass result, which holds for Fano toric Kahler-Einstein manifolds.

Joint work with Peng Zhou and Pierre Flurin.

### Partial orders on contactomorphism groups and their Lie algebras

Eliashberg, Kim and Polterovich constructed nontrivial partial orders on contactomorphism groups of certain contact manifolds. After recalling their results, the subject of this talk will be the remnants of these partial orders on the orbits of the coadjoint action on their Lie algebras.

### Gauge theory for string algebroids

In this talk I will overview recent joint work with Roberto Rubio and Carl Tipler in arXiv:2004.11399. We introduce a moment map picture for string algebroids, a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329. An interesting feature of our construction is that the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of the moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions. Assuming a technical condition, we prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system.

### Geometry of symplectic rational surfaces

Current organizers: Rosa Sena Dias, José Mourão.