Factorization Homology and Low-Dimensional Topology

近几十年来，在低维拓扑学和数学物理（特别是量子场论）的界面上，研究发展呈爆炸式增长。这些发展主要围绕着拓扑不变量：代数对象对拓扑对象的赋值。例如，欧几里得三维空间中的每个连杆（即打结曲线）可以被赋予一个多项式，称为它的琼斯多项式，以及一个向量空间，称为它的Khovanov同调。作为另一个例子，每个3-流形（即三维空间）可以分配一个称为其Heegaard花同调的向量空间。通常，这些不变量是通过首先做出任意选择来定义的（例如，在欧几里得3空间中表示所选链路的链接图，或表示所选3流形的Heegaard图）；为了得到拓扑不变量，必须证明结果与这个选择无关。因式同调是一种定义拓扑不变量的工具，它不需要任何任意选择，它带来了如下所述的许多优点。本项目旨在进一步发展分解同调理论，并将其应用于研究低维物理拓扑中的不变量，如下所述。此外，它还支持早期职业数学家的培训，组织生态可持续会议，并以传播数学理解的喜悦和兴奋为目标的公众参与。因式同调是积分的一种分类形式：普通积分求和数，因式同调求和向量空间或链复形。更准确地说，因式同调在n个流形上集成（丰富的）n个范畴，给出了n个流形的不变量，该不变量遵循TQFT中出现的局部到全局原则。鉴于其良好的定义性和同伦相干的泛函性（包括微分同态的连续作用），它在状态和TQFT中起着类似于奇异同调在细胞同调中的作用。此外，一个轻微的修改有望给出3流形中链接的不变量，例如Witten—Reshetikhin—Turaev不变量（在3流形的skein模块中取值）。本项目还寻求将分解同调应用于Khovanov同调和其他链接同调理论的研究。除了使它们明确定义之外，这种连接将进一步赋予它们对任意3流形中的链接的扩展和对协点的同伦相干功能。由此产生的对称性导致了与算术对象的推测关系，如环切分光谱和拓扑卡地亚模块，这也是本项目试图探索的。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。

Recent decades have seen an explosion of research developments at the interface of low-dimensional topology and mathematical physics (particularly quantum field theory). These developments largely center around topological invariants: assignments of algebraic objects to topological objects. For example, each link (i.e. knotted curve) in Euclidean 3-space can be assigned a polynomial called its Jones polynomial, as well as a vector space called its Khovanov homology. As another example, each 3-manifold (i.e. 3-dimensional space) can be assigned a vector space called its Heegaard Floer homology. Typically, these invariants are defined by first making an arbitrary choice (e.g. a link diagram representing the chosen link in Euclidean 3-space, or a Heegaard diagram representing the chosen 3-manifold); in order to have a topological invariant, one must then show that the result is independent of this choice. Factorization homology is a tool for defining topological invariants that does not require any arbitrary choices, which brings a number of advantages as discussed below. The present project seeks to further develop the theory of factorization homology, and to apply it to study invariants in low-dimensional topology of physical interest as also discussed below. It furthermore supports the training of early career mathematicians, the organization of ecologically sustainable conferences, and public engagement with the goal of disseminating the joy and excitement of mathematical understanding.Factorization homology is a categorified form of integration: whereas ordinary integration sums numbers, factorization homology sums vector spaces or chain complexes. More precisely, factorization homology integrates (enriched) n-categories over n-manifolds, giving an invariant of n-manifolds that adheres to the local-to-global principles appearing in TQFT. Given its well-definedness and homotopy-coherent functoriality (including a continuous action of diffeomorphisms), it plays an analogous role for state-sum TQFT as singular homology plays for cellular homology. Moreover, a slight modification is expected to give an invariant of links in 3-manifolds, e.g. the Witten--Reshetikhin--Turaev invariant (valued in the skein module of the 3-manifold). The present project also seeks to apply factorization homology to the study of Khovanov homology and other link homology theories. Beyond making them manifestly well-defined, such a connection would further endow them with extensions to links in arbitrary 3-manifolds and with homotopy-coherent functoriality for cobordisms. The symmetries that would result lead to conjectural relationships with arithmetic objects such as cyclotomic spectra and topological Cartier modules, which the present project also seeks to explore.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.