I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

/pdf/deformation-quantization-of-poisson-manifolds-4tp4kn8ari.pdf

Deformation Quantization of Poisson Manifolds

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Introduction to lie algebras and representation theory

The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.

/pdf/the-ricci-flow-an-introduction-34asb07sd2.pdf

The Ricci Flow: An Introduction

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Operads in algebra, topology, and physics

We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two-colored operad and characterize its algebras via morphisms of Frobenius algebras, recovering some previously known results of Kaufmann, Penner and others. As an important auxiliary tool we characterize inclusions of cyclic operads that induce inclusions of their modular completions.

Open-closed modular operads, Cardy condition and string field theory

We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras and their diagrams.

Universal Deformation Formulas

This note was originated many years ago as my reaction to questions of several people how free strongly homotopy algebras can be described and what can be said about the structure of the universal enveloping A(m)-algebra of an L(m)-algebra, and then circulated as a "personal communication." I must honestly admit that it contains no really deep result and that everything I did was that I expanded definitions and formulated a couple of statements with more or less obvious proofs.

Free Homotopy Algebras

By an odd structure we mean an algebraic structure in the category of graded vector spaces whose structure operations have odd degrees. Particularly important are odd modular operads which appear as Feynman transforms of modular operads and, as such, describe some structures of string field theory. 
We will explain how odd structures are affected by the choice of the monoidal structure of the underlying category. We will then present two `natural' and `canonical' constructions of an odd modular endomorphism operad leading to different results, only one being correct. This contradicts the generally accepted belief that the systematic use of the Koszul sign convention leads to correct signs.

Odd structures are odd

We develop a new formalism for the Quantum Master Equation $\Delta e^{S/\hbar} = 0$ and the category of ${\rm IBL}_\infty$-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of ${\rm IBL}_\infty$-algebras and ${\rm L}_\infty$-algebras, and on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.

Martin Markl

Papers

Open-closed modular operads, Cardy condition and string field theory

Universal Deformation Formulas

Free Homotopy Algebras

Odd structures are odd

The MV formalism for ${\rm IBL}_\infty$- and ${\rm BV}_\infty$-algebras