# Geometria em Lisboa Seminar

## Planned sessions

### $G_2$-monopoles (a summary)

This talk is aimed at reviewing what is known about $G_2$-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of $G_2$-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in $G_2$-geometry.

### Localizing the Donaldson-Futaki invariant

We will see how to represent the Donaldson-Futaki invariant as an intersection of equivariant closed forms. We will use it to express this invariant as the intersection on some specific subvarieties of the central fibre of the test configuration. As an application we provide a proof that for Kähler orbifolds the Donaldson-Futaki invariant is the Futaki invariant of the central fiber.

### On the space of Kähler metrics

Inspired by the celebrated $C^0, C^2$ and $C^3$ a priori estimate of Calabi, Yau and others on Kähler Einstein metrics, we will present an expository report of a priori estimates on the constant scalar curvature Kähler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the properness conjecture on Mabuchi energy functional.

This is a joint work with Cheng JingRui.

### New consequences of convexity beyond dynamical convexity

Current organizers: Rosa Sena Dias, José Mourão.