January 18, 2022

**Title:** *On the cohomology of moduli spaces of stable pointed curves: genus 4 and small n, and the vanishing of H^7 and H^9*

**Speaker:** Carel Faber

After an introduction to moduli spaces of curves, I will discuss joint work with Jonas BergstrÃ¶m and Sam Payne. For n at most 3, we determine the relevant terms of the point count of the moduli space M_{4,n} of smooth n-pointed curves of genus 4 over finite fields (S_n-equivariantly). Using several earlier results, we then determine the cohomology of the moduli space overline{M}_{4,n} of stable n-pointed curves of genus 4 and the full count for M_{4,n} and we deduce the mentioned vanishing for all g and n