Algebraic Structures in String Topology

本项目的目标是了解背景空间中弦相互作用的一般结构及其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。字符串、路径和循环的相互作用在数学和科学中无处不在。这些范围从流体动力学中可观察到的现象（流体中的涡旋环聚集在一起成为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。弦拓扑提出了一个数学模型来研究这些相互作用的操作，这些操作定义为在流形中随时间演变的相交、重新连接和切断弦（封闭曲线）。对弦拓扑结构进行严谨、完整的描述是本课题的目标之一，也将为物理理论提供坚实的基础。此外，受物理启发的弦拓扑学理论被证明为数学中的理论问题提供了信息：通过弦探测空间，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先前工作的基础上，该项目提出通过分解或离散底层空间为单元并使用代数拓扑和同调代数（两个发展良好的纯数学活跃领域）的技术获得的可处理模型来代数化弦拓扑。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组成部分，重点是促进多层次的数学活动和获取。这包括研究生培训，组织夏季研讨会和会议，将来自各个领域的研究人员聚集在一起，以及支持PI所在机构的定期研讨会。在更多的技术细节中，该项目旨在研究链级字符串拓扑，重点关注对底层流形的同伦类型以外的几何结构敏感的操作。特别是，PI提出构造一个同伦相干结构，提升了最初定义在流形上自由环空间相对于常数环的同调上的Goresky-Hingston环协代数（及其S^1对称Lie协托版本）。这种结构的构建将使用适当细化的poincar<s:1>对偶性和交理论在精细三角化流形上的链水平，以及PI先前的工作中使用Hochschild同调理论和Koszul对偶理论开发的非单连通流形环空间的代数模型理论。这些模型将足够透明，以揭示精确的几何成分，这些成分对于构建字符串拓扑结构的高层结构的连贯层次是必要的。这种链级操作的层次结构将提供可计算的和潜在的新流形不变量的丰富来源。将探讨与辛几何、同调镜像对称和量子化理论的联系。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。

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.