Titles and abstracts

Assar Andersson

Title: Complex of oriented graphs with precisely one target

Abstarct: I will show that there is an injection H(GC^or_d) -> H(GC^or_d/(GC^or_{> 2t}, Δ_closure(GC^{or, > 4}_d), where GC^or is the Kontsevich oriented graph complex, GC^or_{> 2t} is the subcomplex spanned by graphs with at least 2 target vertices, and Δ_closure(GC^{or, > 4}_d) is the differential closure of the subspace spanned by graphs with at least one 4-valent vertex or a 3-valent source or target vertex.

It follows that every cohomology class of degree n in GC^or₃ contains some terms with precisely one bi-valent target vertex and n+2 bi-valent source vertices, joined together by 3-valent vertices that are not sources or targets.

 

Vladimir Dotsenko

Title: Homotopy type of the moduli space of stable rational complex curves

Abstract: I shall show that the rational cohomology of the moduli space of stable rational complex curves is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way. Time permitting, I shall talk about a few classes of spaces for which similar results are available, and a few other conjectural classes of spaces like that.

 

Alexey Kalugin

Title: On a conjectural motive associated with a stratified variety.

Abstract: I will describe Betti and de Rham realizations of a conjectural motivic structure associated with a stratified variety. This motive controls quantizations of Lie bialgebras and connected with a motivic fundamental group.

 

Anton Khoroshkin

Title: Cacti groups, real locus of Deligne-Mumford compactification M_0,n+1.

Abstract: The real locus of Deligne-Mumford compactification M_0,n+1(R) is known to be the Eilenberg-Maclane space of the so called pure cacti group.

This group has a lot of common properties with pure braid group. In particular, (pure) cacti group acts naturally on tensor products of representations of quantum groups U_q(g). This action has a well-defined simple combinatorial limit for q -> 0.

I will report on old and new results on the (pure) cacti group.

In particular, I will explain how the language of operads provides the description of the rational homotopy type of M_0,n+1(R).

The talk is based on the joint work with Thomas Willwacher arXiv:1905.04499.

 

Speaker: Andrey Lazaraev

Title: Strong homotopies of dg algebras: a global approach to deformation theory.

Abstract: The modern understanding of deformation theory is based on differential graded (dg) Koszul duality as developed by Hinich, Lefevre and Positselski. Any dg Lie algebra gives rise to a deformation functor defined on the homotopy category of commutative differential graded local pseudocompact algebras and is representable in this category. One may attempt to extend the deformation functor to not necessarily local (dg) commutative algebras as a way of classifying objects that are not infinitesimally close to a given one; however determining the gauge action and thus, the true moduli functor is problematic in this approach.

We show that, surprisingly, this problem can be overcome if the deformation problem is associated with a (dg) associative algebra. In this context the relation of a gauge equivalence of Maurer-Cartan elements can be usefully relaxed to a homotopy gauge equivalence, a new notion introduced in a joint work with J. Holstein and J. Chuang for the purpose of developing a higher version of Riemann-Hilbert correspondence. With this relaxation, the global deformation functor is suitably representable up to homotopy.

 

Francois Petit

Title: Fourier-Mukai transforms for deformation quantization modules

Abstract: Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves. This is a joint work with David Gepner.

 

Sergey Shadrin

Title: Brick manifolds as noncommutative M_0,1+n.

Abstract: Brick manifolds are some very special quiver varieties (that can also be described as toric varieties of Loday's associahedra, or some De Concini-Procesi wonderful models). They form a nonsymmetric topological operad that exhibits all the remarkable algebraic and geometric features of the operad of the Deligne-Mumford compactifications of the moduli spaces of genus 0 curves with marked points, with a difference that all the related concepts lose commutativity. I'll try to make a survey of that and in particular to answer the following question: what is a noncommutative hypercommutative algebra?

It is a joint work with Vladimir Dotsenko and Bruno Vallette.

 

Marko Zivkovic

Title: Calculation of complete hairy graph cohomology with the connecting differential

Abstract: On the hairy graph complex HGC_-1,1 there is an extra differential Δ that connects a hair to another edge, transforming the hair into an edge. Δ commutes with the standard differential δ that splits a vertex, so it can be used to better understand standard cohomology. I will explain how to calculate the cohomology of ( HGC_-1,1,Δ) using some theory and computer calculations. The similar procedure is expected to work with common sub-complexes, namely those spanned by connected graphs, graphs with vertices at least 2- or 3-valent, etc.

Presentation, Results for all graphs, Results for at least 1-valent vertices, Results for connected graphs.