**Title:** Finiteness of Teichmüller curve in genus three

**Speaker:** Matthew Bainbridge

**Speaker Info:** Indiana University

**Brief Description:**

**Special Note**:

**Abstract:**

There are many closed geodesics in the moduli space M_g of genus g Riemann surfaces, and each of these geodesics can be extended to an isometrically immersed hyperbolic plane in M_g. Usually, these hyperbolic planes are dense in M_g, but very rarely one may cover an algebraic curve. Such isometrically immersed algebraic curves in M_g are called Teichmüller curves. There are infinitely many "primitive" Teichmüller curves in M_2, discovered by McMullen and Calta. In further work, McMullen classified Teichmüller curves in M_2, but the situation in higher genus remained mysterious.In this talk, I'll discuss recent work with Martin Moller and Philipp Habegger showing that there are only finitely many "algebraicallly primitive" Teichmüller curves in M_3. The proof uses ideas from algebraic geometry, number theory, and conformal geometry.

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