Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

Collection Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

Organizer(s) Erik PANZER (University of Oxford) & Karen YEATS (University of Waterloo)

Date(s) 16/11/2020 - 20/11/2020

linked URL https://indico.math.cnrs.fr/event/4834/

00:00:00 / 00:00:00

25 28

The Euler Characteristic of $Out(F_n)$ and the Hopf Algebra of Graphs

By Michael Borinsky

rational Euler characteristic

In their 1986 work, Harer and Zagier gave an expression for the Euler characteristic of the moduli space of curves, $M_{gn}$, or equivalently the mapping class group of a surface. Recently, in joint work with Karen Vogtmann, we performed a similar analysis for $Out(F_n)$, the outer automorphism group of the free group, or equivalently the moduli space of graphs. This analysis settles a 1987 conjecture on the Euler characteristic and indicates the existence of large amounts of homology in odd dimensions for $Out(F_n)$. I will illustrate these results and explain how the Hopf algebra of graphs, based on the works of Kreimer, played a key role to transform a simplified version of Harer and Zagier's argument, due to Kontsevich and Penner, from $M_{gn}$ to $Out(F_n)$. This combined technique can be interpreted as a renormalized topological field theory. I will also report on more recent results on the integer Euler characteristic of $Out(F_n)$.

Information about the video

Date of recording 20/11/2020
Date of publication 30/11/2020
Institution IHES
Language English
Audience Researchers
Format MP4

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