Adrian Diaconu : Equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

April 27, 2018, 11 a.m.

In

SEO 612

Abstract

Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this talk, I will discuss an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ $\textrm{max}\{0, 3 - 2g' \} \le n' \le 2(g - g') + n.$ This answers a question posed by Getzler and Kapranov by making their integral representation of the characteristic of the modular operad $\mathbb{M}\mathcal{V}$ effective. I will also discuss some applications.

Adrian Diaconu will also speak in the Departmental Colloquium on April 27, 2018.