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The Intricate Maze of Graph Complexess

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Vasily Dolgushev
Temple University
Sam, 2016-03-19 10:00 - 11:00
MPIM Lecture Hall

In the paper ``Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes $\mathcal{GC}_{\text{Com}}$, $\mathcal{GC}_{\text{Lie}}$ and $\mathcal{GC}_{\text{As}}$ ``assembled from'' graphs with some additional structures. The graph complex $\mathcal{GC}_{\text{Com}}$ (resp. $\mathcal{GC}_{\text{Lie}}$, $\mathcal{GC}_{\text{As}}$) is related to the operad $\text{Com}$ (resp. $\text{Lie}$, $\text{As}$) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes $\mathcal{GC}_{\text{Com}}$, $\mathcal{GC}_{\text{Lie}}$ and $\mathcal{GC}_{\text{As}}$ (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms $\mathrm{Out}(F_r)$ of the free group $F_r$ on $r$ generators, the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.

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