Homotopy Theory of Algebras and Its Applications

该奖项支持在数学和物理之间进行的研究。拟议的项目涉及包括代数和几何在内的几个领域的交叉。这些问题的起源源于镜像对称，这是理论物理学家最近发现的一种有趣的现象。因此，拟议项目的解决方案将加强我们对有趣的、深刻的数学和来自物理学的可应用的、令人兴奋的想法之间的相互作用的理解。部分资金还将用于支持研究生参与建议的项目，以及参加会议。建议的项目使用代数的同伦理论来研究范畴镜像对称、模空间和导数几何。研究人员计划致力于：(1)L-无限空间的Whitehed型定理，它同时推广了经典的逆函数定理和L-无限代数的Whitehead定理；(2)证明了PI以前在镜像构造方面所做的一些基本性质，如它的Calabi-Yau性质，单调性的消失，紧致化；(3)定义紧光滑Calabi-Yau A-无穷范畴的亏格零Gromov-Witten不变量的一般方法。这个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智力优势和更广泛的影响审查标准进行评估，认为值得支持。

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.