Modern Mathematics Lecture Series | Kähler-Yang-Mills equations and gravitating vortices

Abstract

The search for canonical metrics on manifolds and vector bundles is one of the most natural problems in geometry. In this talk, I will intruduce a system of coupled differential equations, known as the Kähler-Yang-Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the Donaldson--Uhlenbeck--Yau theorem for bundles and the Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics, intertwine the curvature of a Hermitian–Yang–Mills connection on the bundle and the scalar curvature of a Kähler metric on the manifold. After this, I will consider special symmetric solutions on a compact Riemann surface known as gravitating vortices.

About the speaker

Oscar Garcia-Prada

Consejo Superior de

Investigaciones Científicas

Oscar García-Prada is a CSIC Research Professor at Instituto de Ciencias Matemáticas— ICMAT. He obtained a D.Phil. (doctorate) in Mathematics at the University of Oxford in 1991, and had postdoctoral appointments at Institut des Hautes Études Scientific (Paris), University of California at Berkeley, and University of Paris-Sud, before holding positions at University Autónoma of Madrid and École Polytéchnique (Paris). In 2002 he joined the Spanish National Research Council (CSIC).

His research interests lie in the interplay of differential and algebraic geometry with differential equations of theoretical physics, more concretely, in the study of moduli spaces and geometric structures. The moduli spaces considered involve objects such as vortices, solutions to general gauge-theoretic equations, Higgs bundles and representations of surface groups and fundamental groups of higher dimensional Kaehler manifolds.

Personal Homepage:

https://www.icmat.es/miembros/garcia-prada/