Foams, Categorification, and Link Homology

低维的拓扑结构和几何结构与稳定的高维范围有很大的不同。低维的一个特征是被称为tqft（拓扑量子场论）的深层结构的存在，许多起源于量子物理学，并应用于凝聚态物质，统计力学和量子场论。首席研究员将研究这种更一般类型的拓扑理论，其中理论已知于无边界的拓扑对象（封闭对象），并常规地扩展到有边界的对象。这些构造证明了链接同调理论的显式组合构造的成果，其中拓扑对象是嵌入在三维空间中的泡沫状二维结构。作者最近证明了这种结构在一维上的半线性版本扩展了所谓的有限状态自动机和规则语言，这是计算机科学中的一个经典课题。这开启了许多推广的可能性，包括探索更一般的语言和拓扑理论之间的联系，以及二维理论和元胞自动机之间的可能关系。进一步研究拓扑理论和相关主题的泡沫和链接同源性将导致低维拓扑和相关领域的丰硕发现。更具体地说，该项目有三个主要目标。第一个主要目标是进一步发展泡沫的理论、评价及其在链接同源性和分类中的应用。泡沫是具有一般奇异性的二维化合物。他们已经证明了GL(N)链同源理论的组合方法的工具，并拥有诱人的连接到轨道的瞬子花同源。PI将进一步发展与泡沫相关的拓扑理论，着眼于技术难题，如嵌入式三价图的Kronheimer-Mrowka同调的计算和寻找该同调的组合对应物。第二个目标是找到几种链接同源理论的方法，包括Cautis， Webster和Qi-Sussan同源，以建立它们的功能并扩展到缠结和缠结协同。许多重要的链接同源理论，包括三阶HOMFLYPT同源、Webster、Cautis和Qi-Sussan同源，都缺少对缠结配合的功能扩展，在大多数情况下，缺少对缠结的相关扩展。PI将开发这些同源理论的新方法来重新定义它们，在必要时修复功能，并将它们扩展到链接协点。第三个目标是理解低维的普遍理论。PI将继续研究拓扑理论的普遍构造，受到最近成功的激励，例如通过具有缺陷的一维拓扑理论解释有限状态自动机和正则语言，并在布尔半环B中取值，其中正则语言和圆形正则语言产生刚性对称单轴B-线性范畴。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。

Topology and geometry in low dimensions differ significantly from those in the stable, high-dimensional, range. One feature of low dimensions is the existence of deep structures known as TQFTs (Topological Quantum Field Theories), many originating in quantum physics and having applications to condensed matter, statistical mechanics and quantum field theory. The Prinicipal Investigator will be studying such topological theories of more general type, where the theory is known on topological objects without boundary (closed objects) and extended canonically to objects with boundary. These constructions proved fruitful for explicit combinatorial construction of link homology theories, where topological objects are foam-like two-dimensional structures embedded in 3-space. The author has recently shown that a semi-linear version of this construction in dimension one extends so-called finite state automata and regular languages, which is a classical subject in computer science. This opens possibility of many generalizations, including exploring connections between more general languages and topological theories and possible relations between two-dimensional theories and cellular automata. Further studies of topological theories and related topics of foams and link homology should lead to fruitful discoveries in low-dimensional topology and related fields.More specifically, the project has three major goals. The first major goal is to further develop the theory of foams, their evaluations and applications in link homology and categorification. Foams are two-dimensional CW-complexes with generic singularities. They have proved instrumental in combinatorial approaches to GL(N) link homology theories and boast tantalizing connections to instanton Floer homology for orbifolds. The PI will further develop topological theories related to foams, with an eye towards technically difficult problems, such as computation of Kronheimer-Mrowka homology of embedded trivalent graphs and finding combinatorial counterpart of that homology. The second goal is to find approaches to several link homology theories, including Cautis, Webster and Qi-Sussan homologies, to establish their functoriality and extend to tangles and tangle cobordisms. A number of important link homology theories, including triply-graded HOMFLYPT homology, Webster, Cautis, and Qi-Sussan homology, are missing a functorial extension to tangle cobordisms and, in most cases, a related extensions to tangles. The PI will develop new approaches to these homology theories to redefine them, repair functoriality where necessary, and extend them to link cobordisms. The third goal is to understand universal theories in low dimensions. The PI will continue studying universal construction of topological theories, motivated by recent successes such as the interpretation of finite state automata and regular languages via one-dimensional topological theories with defects and taking values in the Boolean semiring B, where a regular language and a circular regular language give rise to a rigid symmetric monoidal B-linear category.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.