The moduli space of rank n metric graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex in degree n all have the same rational cohomology. We determine the asymptotic behavior of the associated Euler characteristic, and thereby prove that the total dimension of this cohomology grows rapidly with n. This is joint work with Michael Borinsky.
The Chillingworth subgroup of the mapping class group of a compact oriented connected surface of genus g with one boundary component is defined as the subgroup of the mapping class group of the surface, whose elements preserve nonsingular vector fields on the surface up to homotopy. We determined the rational abelianization of the Chillingworth subgroup as a full mapping class group module....
The aim of this talk is to introduce a natural lifting, to the level of smooth real differential forms, of the systems of tautological rings in the real-valued cohomology of the moduli spaces of marked compact Riemann surfaces. The system of rings of tautological forms can be described as the smallest system of forms that is closed under all tautological pullbacks and submersions, and contains...
Questions about the growth rate of zeta functions and L-functions are a central topic in analytic number theory. In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith posed a conjecture on the asymptotics of moments of quadratic L-functions. While these sorts of problems originate as questions about number fields, they have a more geometric version when posed over function fields in...
By combining Borel's stability and vanishing theorem for the stable cohomology of GL(n,Z) with coefficients in algebraic GL(n,Z)-representations and the Hochschild-Serre spectral sequence, we study the stable twisted cohomology of the automorphism group Aut(F_n) of the free group F_n of rank n and the stable rational cohomology of the IA-automorphism group IA_n of F_n. We propose a conjectural...
A homology cylinder is a 3-manifold that is homologically the product of a surface and an interval.
In this talk, we introduce the Reidemeister-Turaev torsion of homology cylinders which takes values in the K_1-group of the I-adic completion of the group ring of the fundamental group of a surface over the rationals, and prove that its reduction by the ideal \hat{I}^{d+1} is a finite-type...
I will explain a geometric argument to construct infinitely many non-zero unstable cohomology classes for the group GL_g(Z), some of which were known or conjectural, and others which are new.
By a deep result of Hain, we know generators and relations of the (relative) Malcev completion of mapping class groups. In the limit where the genus goes to infinity, there is a description of that Lie algebra as the cohomology of a certain graph complex (closely related to higher genus Grothendieck-Teichmüller Lie algebras). By computing the cohomology of the Koszul dual graph complex, one...
Automorphism groups of free groups are related to mapping class groups and gives rise to nice subgroups IA, analogous to Torelli groups, with two fundamental filtrations: the Johnson-Andreadakis one, and its lower central series. All these objects carry a deep structure and are very hard to approach. In this talk, we will give a survey of what the use of functor categories can bring to their...
The stable cohomology of Aut(F_n) has been studied by several authors. Galatius proved that the stable cohomology groups with coefficients in Q are trivial. With coefficients in tensor powers of H=H_1(F_n,Q), or of its dual H*, the stable cohomology groups were independently computed by Djament and Vespa (using functor homology methods) and by Randal-Williams (by extending the methods of...
There are many different graph complexes which are often used in applications, e.g. in the deformation quantization theories, in the algebraic topology, in the theory of moduli spaces of algebraic curves, in the Lie theory, etc. Examples include the Kontsevich graph complex, its directed version, the complex of oriented graphs, the one of sourced graphs, the one of sourced-and-targeted...
In my talk I will explain several basic notions and properties developed in Ecalle's mould theory and Sauzin's dimould theory. By introducing the balance map for dimoulds and extending the notion of Zag, I will explain how associators are reformulated in terms of mould theory. My talk is based on my joint work with M.Hirose and N.Komiyama.
By interpreting the pentagon equation defining the Grothendieck-Teichm¥"uller Lie algebra as a condition on the normal form of certain elements in the 5-strand Lie braid algebra, we show how the double shuffle defining equations arise naturally in grt. This gives an intrinsic and elementary proof of the injection of grt into double shuffle first shown by H. Furusho using double polylogarithms.
We prove that homological stability with respect to connected sums of S^2×S^2 fails for moduli spaces BDiff(X) of simply-connected closed 4-manifolds X. This makes a striking contrast with all other even dimensions, where analogous stability has been established by Harer in dimension 2 and by Galatius and Randal-Williams in dimension higher than 4. The proof of the above result is based on a...
