Algebraic Structures in String Topology

这个项目的目标是了解背景空间中弦相互作用的一般结构，它在几何和数学物理中的意义，并使用代数模型进行显式计算。弦、路和循环的相互作用在整个数学和科学中无处不在。这些现象从流体力学中的可观察现象(流体中的涡环聚集成一个新的环或自相交并分裂成多个环)到理论物理领域中出现的模式，如弦理论和量子场论。弦拓扑提出了一种数学模型来研究这些相互作用，这些运算是通过在流形中相交、重新连接和切割随时间演变的弦(闭合曲线)来定义的。对弦拓扑结构的严格和完整的描述也将为物理理论提供坚实的基础，这是拟议项目的目标之一。此外，受到物理启发的弦拓扑学理论被证明是数学中的理论问题：通过弦探索空间并研究所有可能的相互作用是如何组织的，也揭示了背景几何的复杂方面。在PI以前工作的基础上，该项目建议通过将底层空间分解或离散成单元并使用代数拓扑和同调代数这两个发展得很好的纯数学活跃领域的技术来获得易处理的模型来代数化字符串拓扑。这些模型将适用于研究纯数学和应用数学以及理论物理中出现的广泛的弦相互作用现象。拟议的项目包括一个广泛的教育部分，侧重于促进数学活动和在多个层次上获得数学知识。这包括研究生培训，组织暑期讲习班和会议，让来自不同领域的研究人员齐聚一堂，并支持在国际和平研究所定期举办研讨会。在更多的技术细节中，这个项目旨在研究链级字符串拓扑，重点是对基础流形同伦类型以外的几何结构敏感的操作。特别是，PI建议构造一个同伦凝聚结构，提升最初定义在流形上自由环空间相对于常环的同调上的Goresky-Hingston环余代数(及其S^1-对称李余括号形式).这种结构的构造将使用精细三角流形上链水平上的Poincaré对偶和交集理论的适当细化，以及PI以前的工作中使用Hochschild同调理论和Koszul对偶理论的技巧发展的非单连通流形的环空间的代数模型理论。这些模型将足够透明，以揭示为弦拓扑构建连贯的高层结构所必需的精确几何成分。这种链级操作的层次结构将提供丰富的可计算的和潜在的新流形不变量来源。本课程将探讨与辛几何、同调镜像对称性和量子化理论的联系。该奖项反映了NSF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，被认为值得支持。

The goal of this project is to understand the general structure of string interactions in a background space, its significance in geometry and mathematical physics, and to carry out explicit computations using algebraic models. Interactions of strings, paths, and loops are ubiquitous throughout mathematics and science. These range from observable phenomena in fluid dynamics (vortex rings in a fluid coming together to become a new ring or self-intersecting and breaking apart into multiple rings) to patterns arising in areas of theoretical physics such as string theory and quantum field theory. String topology proposes a mathematical model to study these interactions in terms of operations defined by intersecting, reconnecting, and cutting strings (closed curves) evolving in time in a manifold. Giving a rigorous and complete description of the structure of string topology, which is one of the aims of the proposed project, will also provide solid foundations for physical theories. Furthermore, the physicially inspired theory of string topology turns out to inform theoretical questions in mathematics: probing a space through strings and studying how all possible interactions are organized also reveals intricate aspects of the background geometry. Building upon previous work of the PI, the project proposes to algebraicize string topology through tractable models obtained by decomposing, or discretizing, the underlying space into cells and using techniques from algebraic topology and homological algebra, two well developed active fields of pure mathematics. These models will be applicable to study a wide range of string interaction phenomena appearing in both pure and applied mathematics as well as in theoretical physics. The proposed project includes a broad educational component focused on fostering mathematical activity and access at multiple levels. This involves graduate student training, organization of summer workshops and conferences that bring together researchers from a wide variety of fields, and the support of periodic seminars at the PI’s institution. In more technical detail, this project aims to study chain-level string topology with a focus on operations that are sensitive to geometric structure beyond the homotopy type of the underlying manifold. In particular, the PI proposes to construct a homotopy coherent structure lifting the Goresky-Hingston loop coalgebra (and its S^1-symmetric Lie cobracket version) originally defined on the homology of the space of free loops on a manifold relative to the constant loops. The construction of such structure will use an appropriate refinement of Poincaré duality and intersection theory at the level of chains on a finely triangulated manifold together with the theory of algebraic models for loop spaces of non-simply connected manifolds developed in previous work of the PI using techniques from Hochschild homology theory and Koszul duality theory. These models will be transparent enough to reveal the precise geometric ingredients that are necessary to construct a coherent hierarchy of higher structures for string topology. This hierarchy of chain-level operations will provide a rich source of computable and potentially new manifold invariants. Connections with symplectic geometry, homological mirror symmetry, and the theory of quantization will be explored.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.