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

拟议的项目有两个主题。首先是对代数几何中自然产生的A-无穷代数的研究。虽然A-无穷代数最初是在拓扑环境中引入的，但它已被证明在理解代数和几何中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。特别令人感兴趣的是对他们的Hochschild理论的研究，该理论支配着A-无穷大代数的变形方式。第二个主题是研究被称为Pandharipande-Thomas不变量的数值不变量的显式例子，这些不变量最近被引入来理解几何和弦论情况下的曲线计数问题。PI建议尝试解释物理学家观察到的表面上无关的空间的不变量之间的某些意想不到的重合。代数几何是对多项式方程的解的几何研究，在过去的几年中有了重大的发展，特别是在它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中的应用，特别是在弦理论中。这个项目将增加我们对代数几何和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.