Factorization Homology and Low-Dimensional Topology

近几十年来，低维拓扑和数学物理(特别是量子场论)之间的研究发展呈现爆炸性增长。这些发展在很大程度上围绕着拓扑不变量：将代数对象分配给拓扑对象。例如，欧几里得3-空间中的每个链环(即纽结曲线)可以被分配一个称为其琼斯多项式的多项式，以及一个称为其Khovanov同调的向量空间。作为另一个例子，每个3-流形(即3-维空间)可以被分配一个称为其Heegaard Floer同调的向量空间。通常，这些不变量是通过首先进行任意选择来定义的(例如，表示欧几里得3-空间中所选链接的链接图，或表示所选3-流形的Heegaard图)；为了具有拓扑不变性，必须证明结果与该选择无关。因式分解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，这带来了以下讨论的许多优点。本项目旨在进一步发展因式分解同调理论，并将其应用于研究物理兴趣的低维拓扑中的不变量，如下所述。此外，它还支持对早期职业数学家的培训，组织生态可持续发展的会议，并以传播喜悦和数学理解的兴奋为目标的公众参与。因式分解同调是积分的一种分类形式：而普通积分和数字，因式分解同调和向量空间或链复合体。更准确地说，因式分解同调在n-流形上整合(丰富)n-范畴，给出了n-流形的一个不变量，它遵循了TQFT中出现的局部到全局的原则。考虑到它的良定性和同伦凝聚函数性(包括微分同态的连续作用)，它对状态和TQFT的作用类似于奇异同调对细胞同调的作用。此外，稍作修改就有望给出三维流形中的链环的不变量，例如Witten-Reshetikhin-Turaev不变量(取值于三维流形的skein模)。本项目还试图将因式分解同调应用到Khovanov同调和其他链同调理论的研究中。除了明确地定义它们之外，这种联系还将进一步赋予它们到任意3-流形中的环的扩张和对上边界的同伦凝聚函数性。这种对称性将导致与算术对象的猜想关系，如割圆谱和拓扑卡地亚模，本项目也试图探索这一点。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。

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.