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

拟议项目有两个主题。 首先是研究代数几何中自然出现的A-无穷代数。虽然最初是在拓扑背景下引入的，但A-无穷代数已经证明对理解代数和几何中出现的结构至关重要，PI建议从代数几何的角度研究它们。 特别感兴趣的是研究他们的Hochschild理论，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，这是最近推出的理解几何和弦理论的情况下的曲线计数问题。 PI提出试图解释物理学家观察到的某些意想不到的巧合之间的不变量的显然无关的spaces. Algeclamptic几何，这是几何研究的解决方案的多项式方程，已经看到在过去几年的重大发展，特别是在其应用方面的其他领域的科学。 特别重要的是应用到现代理论物理，特别是在弦理论。 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.