The moduli space M_g of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of M_g is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of M_g called the tautological ring. The definition of the tautological ring was later extended to the compactification M_g-bar and the moduli spaces with marked points M_{g,n}-bar. While the full cohomology ring of M_{g,n}-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups H^k(M_{g,n}-bar) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne.

This lecture is part of the Beatrice Yormark Distinguished Lecture Series.

You can learn more about Assistant Professor Hannah Larson here: https://math.berkeley.edu/~hlarson/