Mathematical Sciences: Higher Operations on Hochschild Cohomology

9402076 Voronov The main objective of this project is to work on and prove the following conjecture of Deligne of the Institute for Advanced Study in Princeton: There exists the natural structure of an algebra over the operad of chains of the little disks operad on the Hochschild complex of an arbitrary associative algebra. The significance of the proposed activity is that it provides the Hochschild complex with very rich algebraic structures extending the bilinear cup product and bracket that define a Gerstenhaber algebra structure on the Hochschild cohomology. Those new structures involve higher multilinear products and brackets. More precisely, the structure extending the bracket is expected to be similar to the homotopy Lie algebra structure, which plays a very important role in different branches of physics and mathematics. Such structures have been used by Stasheff and May in their study of loop spaces, Beilinson and Ginzburg and Hinich and Schechtman in the study of deformation theory of algebraic varieties and vector bundles, and by Kontsevich in his study of knot invariants. Physicists Witten and Zwiebach have effectively used the homotopy Lie structure in string theory. The conjecture itself may be reformulated as the existence of a canonical string theory associated to every associative algebra. The conjecture indicates deep connections between algebra and complex analysis. Connections between different branches of mathematics are known to create a lot of excitement and lead to most important discoveries in mathematics. The best recent example may be the Shimura-Taniyama conjecture, which establishes a bridge between number theory and analysis. The recent work of Wiles towards the Shimura-Taniyama conjecture has had many important implications, including his proof of Fermat's Last Theorem. ***