Character varieties and generalizations – Vargen

The study of moduli spaces of G-local systems on a smooth complex algebraic curve (possibly with punctures) and its various incarnations (moduli spaces of Higgs bundles or moduli spaces of principal G- bundles with flat holomorphic connections) is a very active area of research in both mathematics and physics. They play a central ro^le in gauge theory, non-abelian Hodge theory, (higher) Teichmu¨ller theory, hyperbolic 3-manifold theory and Chern-Simons theory. More recently it was shown that their geometry plays also a fundamental ro^le in the theory of integral orbitals (fundamental lemma), geometric Langlands program, representation theory of finite Lie groups, Macdonald polynomials, quiver varieties, DT-invariants from recent Kontsevich-Soibelman theory, the representation theory of double affine Hecke algebras, Hall algebras of a curve and also in the recent work of Deligne-Flicker on the counting of smooth sheaves on curves in positive characteristic. One of the main targets of this ANR project is to understand various connections between these very exciting new developments through the study of the cohomology of these moduli spaces. Another vital aim is to continue to investigate problems in larger but closely related areas such as Satake equivalence or various aspects of the geometric Langlands program.