Foams, Categorification, and Link Homology

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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.