Homotopy Theory of Algebras and Its Applications

代数同伦论及其应用

• 批准号: 1801806
• 负责人: Zhenbo Qin
• 金额: $ 15万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801806&HistoricalAwards=false
• 关键词: Homotopy; Theory; Algebras; Its; Applications

This award supports research at the interface between mathematics and physics. The proposed projects lie at the intersection of several areas including algebra and geometry. The origin of these problems is from mirror symmetry, a recent and fascinating phenomenon discovered by theoretical physicists. As such, solutions of the proposed projects will enhance our understanding of the interaction between interesting, deep mathematics and applicable, exciting ideas from physics. Part of the funding will also be used to support graduate students to work on the proposed projects, and to travel to conferences.The proposed projects use the homotopy theory of algebras to study categorical mirror symmetry, moduli spaces, and derived geometry. The investigator plans to work on: (1) a Whitehead type theorem for L-infinity spaces which simultaneously generalizes the classical inverse function theorem and the Whitehead theorem of L-infinity algebras; (2) proving some essential properties of the PI's previous work on the mirror construction, such as its Calabi-Yau property, vanishing of monodromy, and compactifications; (3) a general approach to define genus zero Gromov-Witten invariants of compact smooth Calabi-Yau A-infinity categories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个奖项支持数学和物理之间的交叉研究。提议的项目位于几个领域的交叉点，包括代数和几何。这些问题的起源来自镜像对称，这是理论物理学家最近发现的一种迷人的现象。因此，提出的项目的解决方案将增强我们对有趣的，深刻的数学和适用的，令人兴奋的物理思想之间相互作用的理解。部分资金还将用于支持研究生在拟议项目中工作，并前往参加会议。提出的项目使用代数的同伦理论来研究范畴镜像对称，模空间和衍生几何。(1)同时推广l -无穷代数的经典反函数定理和Whitehead定理的l -无穷空间的Whitehead型定理；(2)证明了PI先前关于镜像构造工作的一些基本性质，如其Calabi-Yau性质、单态消失和紧化；(3)定义紧光滑Calabi-Yau无穷范畴的格零Gromov-Witten不变量的一般方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。