Puzzles of homotopy algebras related to deformation theory

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To every classical algebra (associative, commutative, Lie) one can assign its homotopy version. In a homotopy version, axioms of the original algebra hold only up to a homotopy and the corresponding homotopy operators are considered as a part of this algebraic structure. These homotopy operators are supposed to satisfy their own coherence laws also up to homotopy and so on. The PI continues his research on homotopy algebras related to Deligne's conjecture on Hochschild complexes and various generalizations of Kontsevich's formality theorem. The PI will continue his work on stable formality quasi-isomorphisms and their homotopy classes. The PI will continue his joint work with D. Tamarkin and B. Tsygan on homotopy calculus structure on Hochschild complexes and apply results to the algebraic index theorem for formal deformations. The PI (jointly with V. Ginzburg) is going to establish a link between homotopy calculus algebras and homotopy BV algebras and apply the results to deformation theory of Calabi-Yau algebras. Finally, the PI is going to investigate a higher categorical structure on homotopy algebras.Homotopy algebras appear in various problems of algebraic topology, algebraic geometry, deformation theory, and mathematical physics. The PI's work on homotopy algebras is motivated by quantization: a process of constructing quantum versions of models of classical mechanics. The project will enhance our understanding of fundamental principles which underpin quantum theory and their links to other branches of mathematics. The PI works with graduate students and postdoctoral researchers, and runs research oriented seminars. The P.I. invites mathematicians from the US and from abroad to speak at the colloquium and at the algebra seminar at Temple University. Talks of invitees give graduate students an opportunity to be exposed to current problems in mathematics.