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Abstract: Character variety is a variety parameterizing representations of the fundamental group of a Riemann surface, or equivalently flat connections, up to equivalence. First examples go back to Fricke and Klein and give rise to the Markov surface. I will talk about a general way to decompose a character variety into simple pieces thus connecting its geometry to interesting combinatorics. Some theorems and conjectures of Hausel, Letellier and Rodriguez-Villegas can be proved using this approach.
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