Departmental Colloquium

Given a genus g surface X with a complex structure, a k-differential is a tensor that looks like f(z)dz^k, where f is complex-analytic in local coordinates z. Strangely, an equivalent definition is that a k-differential is a collection of Euclidean triangles in the plane whose sides have been glued using only compositions of translation and rotations by multiples of 2pi/k. The zeros and poles of a k-differential have orders that sum to k(2g-2). So the moduli space of k-differentials (i.e. a suitably topologized set of all k-differentials) is stratified by specifying an integer partition of k(2g-2) that corresponds to the orders of poles and zeros.

But the topology of these strata is mysterious. So mysterious, in fact, that, in the k=1 case with poles prohibited, the components were only determined by Kontsevich and Zorich in 2003! I’ll describe work with Juliet Aygun determining the components in all cases and that shines some light on their fundamental groups, which is the subject of a conjecture of Kontsevich and which has attracted much recent scrutiny. Time permitting, I’ll connect the story to dynamical results, for systems on both surfaces and strata, and to the groundbreaking work of Eskin and Mirzakhani.