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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 $\bar{M}_g$ and the moduli spaces with marked points $\bar{M}_{g,n}$. While the full cohomology ring of $\bar{M}_{g,n}$ 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 $\bar{M}_{g,n}$, particularly regarding when they are tautological or not. This is joint work with Samir Canning, Sam Payne, and Thomas Willwacher.

There will be a joint reception at from 4pm to 4:30pm in the MSB lobby. The talk will also be simulcast in the 1147 MSB Zoom room; see the e-mail announcement.