Higher algebraic structures, Deligne's conjectures and formality theorems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首先，与M。巴塔宁和私家侦探D·塔玛金将证明Hochschild链的Deligne猜想第二，与D. Tamarkin和B。Tsygan那个私家侦探将证明圆柱上的小圆盘的形式。第三，与D. Tamarkin和B。Tsygan那个私家侦探证明了Batalin-Vilkovisky代数的运算数和微积分的运算数都是Koszul。最后私家侦探研究了Hochschild上链复形上的同伦结合代数结构与该复形上的同伦Gerstenhaber代数结构的对应关系。这些结构的调查是出于各种问题，从代数几何，代数拓扑和数学物理。Hochschild链的Deligne猜想和圆柱上的小圆盘运算的形式猜想是Tamarkin-Tsygan计划的两个剩余的据点，该计划由D.Tamarkin在2000年的Moshe Flato纪念会议上概述。尽管有各种各样的人的努力，这些障碍仍然是开放的。再加上P.I.与D.Tamarkin和B。齐根证明这些命题将完成这一计划。拟议中的研究涉及相当复杂的技术，从不同领域的数学和结果也将影响各种传统领域的调查。这项研究可以应用于变形量子化，这是量子物理数学的基础。

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.