Higher algebraic structures, Deligne's conjectures and formality theorems

高等代数结构、德利涅猜想和形式定理

• 批准号: 0856196
• 负责人: John Baez
• 金额: $ 11万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856196&HistoricalAwards=false
• 关键词: Higher; algebraic; structures; Deligne; conjectures

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.

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，P.I.将与M.Batanin和D.Tamarkin一起证明关于Hochschild链的Deligne猜想。其次，P.I.将与D.Tamarkin和B.Tsygan一起，证明圆柱体上的小圆盘歌剧的正式性。第三，P.I.将与D.Tamarkin和B.Tsygan一起证明巴达林-维尔科维斯基代数的歌剧和微积分的歌剧是Koszul。最后，P.I.将研究Hochschild余链复形上的同伦结合代数结构，它对应于这个复形上的同伦Gertenhaber代数结构。更高的代数结构，如同伦代数或更高的算子，或更高的范畴，在现代数学中扮演着重要的角色。对这些结构的研究是由代数几何、代数拓扑学和数学物理提出的各种问题推动的。对Hochschild链的Deligne猜想和对圆柱体上的小圆盘歌剧的形式猜想是Tamarkin-Tsygan计划剩下的两个据点，该计划由D.Tamarkin在2000年摩西·弗拉托纪念会议上概述。尽管各方都在努力，但这些猜测仍然悬而未决。再加上P.I.与D.Tamarkin和B.Tsygan最近的联合结果，这些猜想的证明将完成这个程序。拟议的研究涉及到来自不同数学领域的相当复杂的技术，其结果也将影响到各种传统的调查领域。这项拟议的研究适用于形变量子化，这是量子物理数学的基础。