Abstract: 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 discuss several results about the cohomology groups of M_{g,n}-bar, particularly regarding when they are tautological or not. This is joint work with Samir Canning, Sam Payne, and Thomas Willwacher.

Bio: Hannah Larson is an Assistant Professor and Clay Research Fellow at University of California, Berkeley. She received her PhD from Stanford University, where she was advised by Ravi Vakil. Her research centers around algebraic curves and their moduli spaces and has been recognized with a 2024 Maryam Mirzakhani New Frontiers Prize.