I will give a tour of some recent ideas which relate: the commutative even graph complex, the moduli space of tropical curves, the cohomology of the general linear group, the Grothendieck-Teichmüller group, MZV's and Feynman integrals.
We explain the construction of a pair of "Betti" and "de Rham" Hopf algebras and a pair of module-coalgebras over this pair, as well as the bitorsors related to both structures (which will be called the "module" and "algebra" stabilizer bitorsors). We show that Racinet's DMR torsor constructed out of the double shuffle and regularization relations between multiple zeta values is essentially...
I will report on a work on the asymptotic growth rate of the top-weight Euler characteristic of the moduli space of curves and on an ongoing joint work with Karen Vogtmann on the topology of Out(Fn). In both cases, graph complexes, which compute the cohomology of the respective spaces, are instrumental. Proofs for the super-exponential asymptotic growth rate of the Euler characteristics in...
I will explain joint work with Oscar Randal-Williams, in which we study Torelli groups of the higher-dimensional analogues of surfaces. This is done by combining the work of Galatius--Randal-Williams on stable moduli spaces of manifolds with Goodwillie--Klein--Weiss embedding calculus. Particular attention will be given to the relationship between our results and graph complexes.
In this short talk I will briefly present admissible integrals, a noncommutative version of the usual integrals on an euclidean space that have to satisfy a property related to integration by part. I will explain all the ingredients in the formula and discuss quickly the existence of such integrals.
For a finite abelian group $G$, Racinet constructed a Lie algebra $\mathfrak{dmr}_0^G$, which for $G=\mu_N$ describes double shuffle and regularisation relations between multiple polylogarithm values specialized to $N^{th}$ roots of unity. Enriquez and Furusho then identified this Lie algebra with the stabilizer Lie algebra $\mathfrak{stab}(Delta^M)$ of a coalgebra $(M, Delta^M)$ appearing in...
For a punctured oriented surface, the isotopy classes of ribbon graph spines on it constitute a cell complex. It can be thought of as a combinatorial model of the Teichmuller space of the surface. The fundamental path groupoid of this cell complex is called the Ptolemy groupoid. In this talk, we discuss various 1-cocycles on the Ptolemy groupoid and the corresponding twisted first cohomology...
This talk is about ongoing joint work with Nancy Hingston and Alexandru Oancea.
I will explain how various puzzles in string topology get resolved in terms of symplectic geometry: Loop space homology and cohomology are merged into a larger space, Rabinowitz Floer homology, which carries the structure of a graded TQFT and satisfies Poincare duality.
Using Thomas Willwacher’s twisting endofunctor, and Kevin Costello’s theory of partially compactified moduli spaces of algebraic curves of arbitrary genus with marked points, we introduce a new dg properad which contains Ezra Getzler’s operad controlling genus zero moduli spaces. We discuss its applications in the theory of moduli spaces $M_g,n$, and in string topology
We construct an algebraic model for the rational homotopy type of Baut(X), the classifying space of fibrations with fiber X, for arbitrary simply connected CW-complexes X. As an application, we express the rational cohomology ring of Baut(X) in terms of cohomology of arithmetic groups and dg Lie algebras. In special cases, this leads to connections to modular forms and to graph complexes in...
The idea behind Grothendieck-Teichmüller theory is to study the absolute Galois group via its actions on (the collection of all) moduli spaces of genus 𝑔 curves. In practice, this is often done by studying an intermediate object: The Grothendieck-Teichmüller group, GT. In this talk, I’ll describe an algebraic gadget built from simple decomposition data of Riemann surfaces. This gadget, called...
After surveying the early developments in profinite
Grothendieck-Teichmüller theory, I will explain how the introduction of
completed curve complexes (as well as other types of complexes) led to
some more recent results, including an important rigidity property which
enables one to define the profinite version of associators.
I will speak about two distantly related topics. The first one is grt(Q) as subspace of the rational homotopy of spaces of long embeddings $R^m -> R^{2n}$, $2n-m>2$.
The second one is invariants of embeddings $f:M^m --> R^n$,$ n-m>2$, in terms of unitrivalent trees that encode the rational homotopy type of the path-component $Emb(M,R^n)_f$ of the embedding space $Emb(M,N)$. The first topic...
Florian Naef showed that the string topology coproduct is not in general invariant under homotopy equivalences, through a lens spaces computation. I will give one point of view on this failure of invariance
