Foams, Categorification, and Link Homology

泡沫、分类和链接同源性

• 批准号: 2204033
• 负责人: Mikhail Khovanov
• 金额: $ 21.88万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204033&HistoricalAwards=false
• 关键词: Foams; Categorification; Link; Homology

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.

低维中的拓扑和几何与稳定的高维范围中的拓扑和几何显著不同。低维的一个特征是存在被称为TQFT(拓扑量子场论)的深层结构，其中许多起源于量子物理，并应用于凝聚态物质、统计力学和量子场论。首席调查员将研究更一般类型的这种拓扑理论，其中该理论已知于无边界的拓扑对象(闭合对象)，并规范地扩展到有边界的对象。这些构造对于链环同调理论的显式组合构造是卓有成效的，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正则语言，这是计算机科学中的一个经典课题。这打开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，以及二维理论和细胞自动机之间的可能关系。对泡沫和链环同调的拓扑学理论及相关主题的进一步研究将在低维拓扑学及相关领域带来丰硕的发现。第一个主要目标是进一步发展泡沫的理论，它们的评价和在链接同源和分类中的应用。泡沫塑料是具有一般奇点的二维CW-络合物。它们在GL(N)链同调理论的组合方法中被证明是有用的，并夸口地与Orbilold的瞬子Floer同调有诱人的联系。PI将进一步发展与泡沫相关的拓扑理论，着眼于技术上的困难问题，如计算嵌入的三价图的Kronheimer-Mrowka同调和寻找该同调的组合对应物。第二个目标是找到几个链同调理论的方法，包括CAUTIS，Webster和QI-Sussan同调，以建立它们的功能，并扩展到Tangles和Tangle Cobordism。许多重要的链同调理论，包括三次分次HOMFLYPT同调、Webster、CAUTIS和QI-Sussan同调，都缺少对纠缠上边同调的函子扩张，并且在大多数情况下，缺少对纠缠的相关扩张。PI将为这些同调理论开发新的方法，以重新定义它们，在必要时修复功能，并将它们扩展到链接密码学。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建，动机是最近的成功，例如通过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，其中正则语言和循环正则语言产生严格的对称单态B-线性范畴。该奖项反映了NSF的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。