Andrey Lazarev

Feynman diagrams and minimal models for operadic algebras.

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevichs ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras.

Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞–, C∞– and L∞–algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C∞–algebras. This generalises and puts in a conceptual framework previous work by Loday and Gerstenhaber–Schack.

Homotopy BV algebras in Poisson geometry

We define and study the degeneration property for

Topological Hochschild cohomology and generalized Morita equivalence

\mathrm {BV}_\infty

Dieudonné modules and p-divisible groups associated with Morava K-theory of Eilenberg-Mac Lane spaces

algebras and show that it implies that the underlying

Graph cohomology classes in the Batalin-Vilkovisky formalism

L_{\infty }

Spaces of Multiplicative Maps between Highly Structured Ring Spectra

algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity

Combinatorics and Formal Geometry of the Maurer–Cartan Equation

\Delta (e^{\xi })=e^{\xi }\Big (\Delta (\xi )+\frac {1}{2}[\xi ,\xi ]\Big )

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish. - See more at: http://www.ams.org/journals/mosc/2013-74-00/S0077-1554-2014-00216-8/#sthash.pBIIcZKa.dpuf

THE STASHEFF MODEL OF A SIMPLY-CONNECTED MANIFOLD AND THE STRING BRACKET

We explore two constructions in homotopy category with alge- braic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley. A central notion of noncommutative ring theory related to Morita equiv- alence is that of central separable or Azumaya algebras. For such an Az- umaya algebra A, its Hochschild cohomology HH � (A, A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topologi- cal Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topo- logical Azumaya R-algebra is the endomorphism R-algebra FR(M, M) of a finite cell R-module. We show that the spectrum of mod 2 topological K- theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum d KU2. This leads to the determi- nation of THH(KU/2, KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A, A) for a noncommutative S-algebra A. AMS Classification 16E40, 18G60, 55P43; 18G15, 55U99