The proposed research concerns low-dimensional topology, specifically geometric classification of topological 4-manifolds and certain aspects of quantum topology. Krushkal proposes an approach to solving the topological 4-dimensional surgery conjecture in the context of the homotopy A-B slice problem. A convenient framework for this approach is provided by topological arbiters, a notion recently introduced in collaboration with M. Freedman. The project includes a number of problems in quantum topology, specific questions concern spin networks, and as a special case a categorified relation between the chromatic polynomial of planar graphs and the Temperley-Lieb algebra. The project also aims to use topological methods to address classification problems for free and certain classes of interacting fermions, an important class of systems in condensed matter physics. Topological invariants have played an important role in recent work on classification of gapped phases of non-interacting fermions by Kitaev and others. K-theory and Bott periodicity turned out to be crucial ingredients in classifying systems with various symmetries. Krushkal plans to work on a number of related problems, including invariants of families of free fermions and extensions to certain classes of interacting systems.
One goal of the project concerns classification of possible large scale shapes of objects that locally look like the Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. This project is aimed at a better understanding of 4-dimensional objects with large fundamental groups, which contain many loops that cannot be contracted. Another part of the project concerns quantum topology, a subject influenced by ideas from both physics and mathematics and connected with statistical mechanics, representation theory, topology, and combinatorics. In particular, the goal is to gain a new conceptual perspective on spin networks, a notion whose origins are in quantum physics, in the recently developed mathematical framework of categorification. The third part of the project explores interdisciplinary connections between Topology and Condensed Matter Physics, focusing on topological insulators. This research, in collabor ation with physicists, will build on recent advances in classification of this remarkable class of physical systems.
The subject of Geometric Topology concerns classification of shapes up to continuous deformations. The focus of this project is on classification of 4-dimensional manifolds (shapes). This dimension is of particular importance due to its significance in Physics, and specifically in this dimension there is a very important distinction between classification up to differentiable deformations and classification up to deformations which are only assumed to be continuous. This project concerned the latter problem for shapes that have large fundamental groups, in other words shapes that have many loops that cannot be contracted. A key question in this subject is known as the "AB slice problem" which analyzes decompositions of the 4-dimensional ball into two parts. The existence of an obstruction to 4-dimensional classification in the context of this problem has been a central open question since the 1980s. The results of this project contributed to a better understanding of several aspects of the problem. The notion of "topological arbiters" was introduced and studied, providing axioms that an obstruction should satisfy. A particularly subtle family of decompositions was investigated, leading to the formulation of a new algebraic invariant. Another part of this project concerned Quantum Topology, a theory of invariants of knots and other 3-dimensional shapes with origins in Quantum Physics. The results of the PI supported by this grant include a categorification of quantum spin networks, a central notion in quantum topology. Applications to invariants of knots and 3-manifolds and to combinatorics have been investigated. While supported by this grant, the PI supervised work of beginning researchers at all levels. Specifically, the PI supervised independent research of undergraduate students in topological combinatorics, as well as research by graduate students and a posdoctoral fellow in quantum topology and higher category theory.
