Cette journée est financée par le PEPS Jeune Chercheuses et Jeunes Chercheurs du CNRS et l'Institut de Mathématiques de Jussieu.

**Exposés de synthèse :**

Joel Fine (Université Libre de Bruxelles)

**An invitation to Einstein 4-manifolds.**

A Riemannian metric g whose Ricci curvature is constant, Ric(g) = Cg, is called “Einstein”. My talk will be a survey of various parts of what is known about Einstein metrics in dimension 4. Topics I hope to cover include: an explanation of why dimension 4 is special; known examples; obstructions; uniqueness (both local and global),… I will end with open questions (ranging from the concrete to speculative!)

Benjamin Sharp (University of Leeds)

**Minimal submanifolds of Riemannian manifolds**

Minimal submanifolds have been an object of mathematical study since the late 18th century (beginning with Lagrange and Euler), and remain at the heart of many problems to this day. A submanifold is minimal if it is "critical” with respect to the area functional - in practice this means that in a sufficiently small neighbourhood of a point, a compactly supported perturbation of the submanifold must increase its area.

I will attempt to cover a variety of topics related to the modern theory of minimal submanifolds, which will touch on; min-max techniques and refined existence results, regularity theory, the relationship between index and topology for hypersurfaces, and some open problems.

Any discussion of my own results will include collaborations with (subsets of) Lucas Ambrozio, Alessandro Carlotto and Reto Buzano.

**Exposé de recherche: **

Hoang-Chinh Lu (Université Paris Sud)

**Geodesic stability in Kahler geometry.**

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics. This is joint work with Tamas Darvas.

Les participants.