College of Liberal Arts & Sciences

# MATHEMATICS COLLOQUIUM - Tian Yang (Stanford University)

**Abstract**: Character varieties of surfaces are central objects in several branches of mathematics, such as geometric topology, differential geometry and mathematical physics. On the character varieties, there is a tautological action of the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will start by reviewing related results toward proving the ergodicity, and introduce two long standing and related conjectures: Goldman’s Conjecture and Bowditch’s Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surface, we disprove Bowditch’s Conjecture by giving counterexamples, yet prove that Goldman’s Conjecture still holds in that case.