Factorization Homology and Low-Dimensional Topology

近几十年来，在低维拓扑学和数学物理学（特别是量子场论）的界面上出现了爆炸式的研究发展。这些发展主要集中在拓扑不变量上：将代数对象赋值给拓扑对象。例如，欧几里得三维空间中的每一个链环（即打结曲线）都可以被赋予一个称为琼斯多项式的多项式，以及一个称为霍瓦诺夫同调的向量空间。作为另一个例子，每个三维流形（即三维空间）可以被分配一个向量空间，称为其Heegaard Floer同调。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-空间中表示所选链接的链接图，或表示所选3-流形的Heegaard图）;为了具有拓扑不变量，必须证明结果与此选择无关。因式分解同调是一种用于定义拓扑不变量的工具，不需要任何任意选择，这带来了如下所述的许多优点。本项目旨在进一步发展因式分解同调理论，并将其应用于研究低维拓扑中的物理不变量，如下所述。它还支持早期职业数学家的培训，生态可持续会议的组织，以及公众参与，目的是传播数学理解的乐趣和兴奋。因式分解同调是积分的一种分类形式：而普通积分求和数，因式分解同调求和向量空间或链复形。更确切地说，因子分解同调在n-流形上集成（丰富）n-范畴，给出了一个遵守TQFT中出现的局部到全局原则的n-流形不变量。由于它的良好定义性和同伦相干函子性（包括一个连续作用的同胚），它对状态和TQFT起着类似的作用，就像奇异同调对细胞同调所起的作用一样。此外，一个轻微的修改，预计将得到一个不变的链接在3-流形，例如维滕-Reshetikhin-Turaev不变量（价值在skein模的3-流形）。本项目还试图将因子分解同调应用于Khovanov同调和其他链同调理论的研究。除了使它们显然是良好定义的，这样的连接将进一步赋予它们任意3-流形中的链接的扩展和同伦相干函子性。对称性，将导致数学关系与算术对象，如分圆光谱和拓扑卡地亚模块，目前的项目也寻求explored.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.