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

广义 A-无穷代数、稳定性结构和 Hochschild 同调

• 批准号: 0901224
• 负责人: Andrei Caldararu
• 金额: $ 21.63万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901224&HistoricalAwards=false
• 关键词: Generalized; infinity; algebras; stability; structures

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.

拟议项目有两个主题。 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 A-无穷代数最初是在拓扑背景下引入的，但事实证明，A-无穷代数对于理解代数和几何中出现的结构至关重要，PI 建议研究它们从代数几何的角度来看。 特别令人感兴趣的是他们的 Hochschild 理论的研究，该理论控制着 A 无穷代数变形的方式。 第二个主题是研究数值不变量的显式示例，称为 Pandharipande-Thomas 不变量，最近引入这些示例来理解几何和弦理论情况下的曲线计数问题。 PI 提议尝试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。代数几何是多项式方程解的几何研究，在过去几年中取得了重大发展，特别是在其应用方面在其他科学领域。 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。 本项目将增加我们对代数几何和物理学中出现的代数结构的一般理解。