Geometry of Hodge Bundles on Riemann Surfaces

Explore complex mathematical concepts in this one-hour lecture focusing on G-Higgs bundles over compact Riemann surfaces and their relationship with Hodge bundles. Delve into the moduli space of G-Higgs bundles and understand the natural C*-action through scaling the Higgs field, with particular attention to fixed points known as Hodge bundles. Learn about the geometric aspects of these mathematical structures, including the Toledo invariant of Hodge bundles and its Arakelov-Milnor bound. Examine the concept of very stable conditions and discover their connections to mirror symmetry and Langlands duality, while understanding the non-abelian Hodge correspondence that establishes a bijection between Hodge bundles and variations of Hodge structure.