Non-abelian Hodge theory relates representations of the fundamental group of a compact Riemann surface X into a Lie group G with holomorphic objects on X known as Higgs bundles, introduced by Hitchin more than 35 years ago. Starting with the case in which G is the circle, and the 19th century Abel-Jacobi's theory, we will move to the case of G=SL(2,R) and the relation to Teichmüller theory. We will then explain how, using Higgs bundles, one can construct generalizations of the classical Teichmüller space for certain Lie groups of higher rank.
Oscar García-Prada is a CSIC Research Professor at Instituto de Ciencias Matemáticas in Madrid. He obtained a D.Phil. in Mathematics at the University of Oxford in 1991, under the supervision of Nigel Hitchin and Simon Donaldson,and had postdoctoral appointments at Institut des Hautes Études Scientific (Paris), University of California at Berkeley, and Université de Paris-Sud, before holding positions at Universidad Autónoma de 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. He participates regularly in public outreach activities on mathematics and its interactions with physics and music, collaborating with main Spanish newspapers, radio and television.