Related papers: Wheeled PROPs, graph complexes and the master equa…
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete…
Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain…
We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\mathcal V}=T(V)\otimes T(V^\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector…
We consider graph complexes with a flow and compute their cohomology. More specifically, we prove that for a PROP generated by a Koszul dioperad, the corresponding graph complex gives a minimal model of the PROP. We also give another proof…
We show that the stable cohomology of automorphism groups of free groups with coefficients obtained by applying Hom(-,-) to tensor powers of the abelianization, is equipped with the structure of a wheeled PROP H. We define another wheeled…
We assign generalised convolutions (resp. traces) to graphs whose edges are decorated by smooth kernels (resp. smoothing operators) on a closed manifold. To do so, we introduce the concept of TraPs (Traces and Permutations), which roughly…
These are expanded notes of author's talk at the ECM 2008 attempting to give an elementary introduction into the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from…
We will consider P-graph complexes, where P is a cyclic operad. P-graph complexes are natural generalizations of Kontsevich's graph complexes -- for P = the operad for associative algebras it is the complex of ribbon graphs, for P = the…
We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps. Continuous…
We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on that wheeled prop faithfully and…
We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves ${\mathcal M}_{g,n}$ (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy…
This is a copy of my bachelor thesis, written under supervision of Sergey Shadrin in 2010. Original abstract: In this thesis we prove that the wheeled Poisson operad is not a wheeled Koszul operad. Chapter 1 introduces operads, the subclass…
A theory of $\infty$-properads is developed, extending both the Joyal-Lurie $\infty$-categories and the Cisinski-Moerdijk-Weiss $\infty$-operads. Every connected wheel-free graph generates a properad, giving rise to the graphical category…
Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor $\phi$. This structure was introduced by Drinfeld…
This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic…
A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…
In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms…
We give a detailed proof of T. Willwacher's theorem arXiv:1009.1654 which links the cohomology of the full graph complex fGC to the cohomology of the deformation complex of the operad GER, governing Gerstenhaber algebras. We also present…
Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master equations, which can be interpreted as…
The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The task is accomplished in three steps. The first step is the construction of a modified cobar complex adapted to a…