The mini-course will be given by Adrian Langer on Higgs bundles and their moduli space with connections to Langlands program:

Lecture 1 Local systems, flat bundles, Higgs bundles Simpson's correspondence (non-abelian Hodge theory) Examples and applications: Rank 1 case and relation to generic vanishing theorems

Lecture 2 Moduli spaces of representations of the fundamental group and hyperkähler structure Multiplicative group action and complex variation of the Hodge structure Hitchin's fibration Examples and applications: rigidity results for representations of the fundamental group

Lecture 3 Spectral covers and generic fibres of Hitchin's fibration Langlands duality as mirror symmetry (after Hausel-Thaddeus) Geometric Langlands conjecture