Puzzles of homotopy algebras related to deformation theory

对于每一个经典代数（结合代数、交换代数、李代数），都可以赋值它的同伦版本。在同伦版本中，原代数的公理只成立于一个同伦，相应的同伦算子被认为是这个代数结构的一部分。这些同伦算子被要求满足它们自己的相干律，并达到同伦等等。PI继续研究与Deligne关于Hochschild复合体的猜想和Kontsevich形式定理的各种推广有关的同伦代数。他将继续研究稳定形式的拟同构及其同伦类。PI将继续与D. Tamarkin和B. Tsygan共同研究Hochschild复合体上的同伦微积分结构，并将结果应用于形式变形的代数指标定理。PI（与V. Ginzburg合作）将建立同伦微积分代数与同伦BV代数之间的联系，并将结果应用于Calabi-Yau代数的变形理论。最后，PI将研究同伦代数上的高范畴结构。同伦代数出现在代数拓扑、代数几何、变形理论和数学物理的各种问题中。PI在同伦代数上的工作是由量子化推动的：一个构建经典力学模型的量子版本的过程。该项目将增强我们对支撑量子理论的基本原理及其与其他数学分支的联系的理解。PI与研究生和博士后研究人员合作，并举办以研究为导向的研讨会。P.I.邀请来自美国和国外的数学家在天普大学的学术讨论会和代数研讨会上发言。受邀者的演讲给了研究生一个接触当前数学问题的机会。

To every classical algebra (associative, commutative, Lie) one can assign its homotopy version. In a homotopy version, axioms of the original algebra hold only up to a homotopy and the corresponding homotopy operators are considered as a part of this algebraic structure. These homotopy operators are supposed to satisfy their own coherence laws also up to homotopy and so on. The PI continues his research on homotopy algebras related to Deligne's conjecture on Hochschild complexes and various generalizations of Kontsevich's formality theorem. The PI will continue his work on stable formality quasi-isomorphisms and their homotopy classes. The PI will continue his joint work with D. Tamarkin and B. Tsygan on homotopy calculus structure on Hochschild complexes and apply results to the algebraic index theorem for formal deformations. The PI (jointly with V. Ginzburg) is going to establish a link between homotopy calculus algebras and homotopy BV algebras and apply the results to deformation theory of Calabi-Yau algebras. Finally, the PI is going to investigate a higher categorical structure on homotopy algebras.Homotopy algebras appear in various problems of algebraic topology, algebraic geometry, deformation theory, and mathematical physics. The PI's work on homotopy algebras is motivated by quantization: a process of constructing quantum versions of models of classical mechanics. The project will enhance our understanding of fundamental principles which underpin quantum theory and their links to other branches of mathematics. The PI works with graduate students and postdoctoral researchers, and runs research oriented seminars. The P.I. invites mathematicians from the US and from abroad to speak at the colloquium and at the algebra seminar at Temple University. Talks of invitees give graduate students an opportunity to be exposed to current problems in mathematics.