Generalized A-infinity algebras, stability structures, and Hochschild homology

拟议的项目有两个主题。第一个是对代数几何中自然出现的a无穷代数的研究。虽然最初是在拓扑学背景下引入的，但a -∞代数已被证明在理解代数和几何中产生的结构方面是必不可少的，PI建议从代数-几何的角度来研究它们。特别有趣的是他们的Hochschild理论的研究，该理论支配着a无穷代数的变形方式。第二个主题是数值不变量的显式例子的研究，被称为Pandharipande-Thomas不变量，它最近被引入来理解几何和弦理论情况下的曲线计数问题。PI计划试图解释物理学家在明显不相关的空间的不变量之间观察到的某些意想不到的巧合。代数几何是对多项式方程解的几何研究，在过去几年中取得了重大进展，特别是在其他科学领域的应用方面。特别重要的是在现代理论物理学中的应用，特别是在弦理论中的应用。本项目将增加我们对代数几何和代数物理中出现的代数结构的一般理解。

There are two themes to the proposed project. The first is the study of A-infinity algebras that arise naturally in algebraic geometry.While originally introduced in topological contexts, A-infinity algebras have proven essential in the understanding of structures arising in both algebra and geometry, and the PI proposes to study them from an algebraic-geometric point of view. Of particular interest is the study of their Hochschild theory, which governs the way A-infinity algebras deform. The second theme is the study of explicit examples of numerical invariants, known as Pandharipande-Thomas invariants, which have been recently introduced to understand counting-of-curves problems in geometric and string-theoretic situation. The PI proposes to try to explain certain unexpected coincidences observed by physicists between invariants of apparently unrelated spaces.Algebraic geometry, which is the geometric study of solutions of polynomial equations, has seen in the last few years major developments, especially in terms of its applications in other fields of science. Of particular importance are applications to modern theoretical physics, in particular in string theory. The present project will increase our general understanding of algebraic structures that arisein algebraic geometry and physics.