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.

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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.

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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

(1995). Strongly homotopy lie algebras. Communications in Algebra: Vol. 23, No. 6, pp. 2147-2161.

Strongly homotopy Lie algebras

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. ...

Models for operads

We prove that strongly homotopy algebras (such as $A_\infty$, $C_\infty$, sh Lie, $B_\infty$, $G_\infty$,...) are homotopically invariant in the category of chain complexes. An important consequence is a rigorous proof that `strongly homotopy structures transfer over chain homotopy equivalences.'

Homotopy algebras are homotopy algebras

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps inducing a homology isomorphism. This approach, naturally arising in string theory, leads us to consider various versions of models. Besides of some applications in topology and homological algebra we show that our theory enables one to prove the existence of homotopy structures on physically relevant spaces. For example, we prove that a closed string-field theory induces a homotopy Lie algebra structure on the space of relative states, which is one of main results of T. Kimura, A. Voronov and J. Stasheff (see [16]). Our theory gives a systematic way to prove statements of this type. The paper is a corrected version of a preprint which began to circulate in March 1994, with some new examples added.

Martin Markl

Papers

Operads in algebra, topology, and physics

Strongly homotopy Lie algebras

Models for operads

Homotopy algebras are homotopy algebras

Models for Operads