Higher algebraic structures, Deligne's conjectures and formality theorems

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project has several goals. First, together with M. Batanin and D.Tamarkin, the P.I. is going to prove the Deligne conjecture for Hochschild chains. Second, together with D. Tamarkin and B. Tsygan, the P.I. is going to prove the formality of the operad of little discs on a cylinder. Third, together with D. Tamarkin and B. Tsygan, the P.I. is going to show that the operad of Batalin-Vilkovisky algebras and the operad of calculi are Koszul. Finally, the P.I. is going to investigate the homotopy associative algebra structure on the Hochschild cochain complex corresponding to the homotopy Gerstenhaber algebra structure on this complex.Higher algebraic structures, such as homotopy algebras or higher operads, or higher categories, play a prominent role in modern mathematics. The investigation of these structures is motivated by various questions from algebraic geometry, algebraic topology and mathematical physics. The Deligne conjecture for Hochschild chains and the formality conjecture for the operad of little discs on a cylinder are the two remaining strongholds of the Tamarkin-Tsygan program which was outlined by D.Tamarkin in 2000 at the Moshe Flato memorial conference. Despite the efforts of various people these conjectures are still open. Together with joint recent results of the P.I. with D.Tamarkin and B. Tsygan the proofs of these conjectures would complete this program. The proposed research involves rather elaborate techniques from different areas of mathematics and the results would also influence various traditional fields of investigation. The proposed research has applications to deformation quantization which underpins the mathematics of quantum physics.