# Gradient Maps associated with actions of Real Reductive Groups

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Joshua Windare

created by daniele on 24 May 2024

12 jun 2024
-- 14:30

Aula 207, DIMAI, Firenze

Seminari di Geometria del Dini

**Abstract.**

This talk is concerned with the Hamiltonian action of a complex reductive group U on a Kahler manifold (Z,ω). The main goal is to investigate a refinement of this setting, namely, the action of a compatible (real) subgroup G of U on a real submanifold X of Z. One can study the geometry of the G-action on X using the associated gradient map and its norm square function, using techniques from convex geometry, complex geometry, Lie group, and Morse theory. In particular, we will discuss the convexity property of the gradient map and the classical Hilbert-Mumford criterion in Geometric Invariant Theory for the G-action on X. The Hilbert-Mumford criterion gives an explicit numerical criterion for testing the (semi, poly)stability of a point in terms of an important G-equivariant function called the maximal weight function. Some results of this talk are joint work with Leonardo Biliotti (University of Parma).