Abstract: The Riemannian world contains two nontrivial classes of homogeneous Einstein manifolds: those with positive scalar curvature, which include compact isotropy irreducible homogeneous spaces, and those with negative scalar curvature. Due to work of Heber, Lauret, Jablonski, Böhm-Lafuente, it is now known that the latter can be described as standard solvmanifolds of Iwasawa type.
In the indefinite world, a new class appears, namely that of Ricci-flat homogeneous manifolds. Even nilmanifolds may admit a Ricci-flat metric --- in fact, there is no known example of a nilpotent Lie group that does not admit a left-invariant Ricci-flat metric.
I will illustrate some constructions of Einstein solvmanifolds and nilmanifolds that are specific to indefinite signature, and discuss the existence of special geometries compatible with such metrics.
This talk includes joint work with Federico A. Rossi and Romeo Segnan Dalmasso.
A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groupsOluwagbenga Joshua Windare Abstract: A systematic treatment of a Hilbert-Mumford criterion for stability theory for an action of a real reductive group G on a real submanifold X of a K¨ ahler manifold Z will be presented. A set of polystable points plays a critical role in the construction of a good quotient of X by the action of G. Under some mild restrictions on the G-action on X, we characterize polystability in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity of the symmetric space G/K, where K is a maximal compact subgroup of G. This is joint work with Leonardo Biliotti.