Mathematical Sciences: Higher Operations on Hochschild Cohomology

小行星9402076 本项目的主要目的是研究并证明普林斯顿高等研究院Deligne的猜想：任意结合代数的Hochschild复形上的小圆盘链运算数上存在代数的自然结构。 所提出的活动的意义在于，它提供了非常丰富的代数结构的Hochschild复杂扩展的双线性杯产品和括号，定义了一个Gerstenhaber代数结构的Hochschild上同调。 这些新结构涉及更高的多线性乘积和括号。 更确切地说，扩展括号的结构被期望类似于同伦李代数结构，它在物理和数学的不同分支中起着非常重要的作用。 这种结构已被用于Stasheff和五月在他们的研究循环空间，贝林森和金斯堡和Hinich和Schechtman在研究变形理论的代数簇和向量丛，并由Kontsevich在他的研究结不变量。 物理学家维滕和茨维巴赫在弦理论中有效地使用了同伦李结构。 这个猜想本身可以被重新表述为与每个结合代数相关联的规范弦理论的存在性。 该猜想表明代数和复分析之间有着深刻的联系。 数学不同分支之间的联系被称为创造了很多令人兴奋的，并导致最重要的发现在数学。 最近最好的例子可能是志村-谷山猜想，它建立了数论和分析之间的桥梁。 怀尔斯最近对志村-谷山猜想的研究有许多重要的意义，包括他对费马大定理的证明。 ***

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. ***