The proposed project concerns several problems in geometric and quantum topology in dimensions 3 and 4. Based on his recent work, the PI proposes an approach to the A-B slice problem, a formulation of the topological 4-dimensional surgery conjecture for non-abelian free groups. In recent years algebraic invariants have been found for key examples of decompositions of the 4-ball, and the goal is to combine them to formulate an obstruction to surgery in the context of the A-B slice problem. The project also includes a number of problems in quantum topology; some of them are motivated by the PI's recent work with Cooper on categorification of the Jones-Wenzl projectors. Specific problems concern algebraic structures underlying categorified spin networks and the colored Jones polynomial, and an approach to categorification of 3-manifold invariants using ideas related to localization. The PI's ongoing collaboration with physicist Fendley is aimed at identifying higher categorical structures in the context of quantum statistical mechanics.
Topology is the study of shapes that locally look like the Euclidean space. Topological study of shapes in dimensions 3 and 4 is particularly important due to connections with physics, and it became apparent in the 1980s that the geometric classification in these "low" dimensions is a substantially more difficult problem than that in dimensions greater than four. At the same time, a wide array of tools and insights from other areas of mathematics and theoretical physics is available for solving topological problems in three and four dimensions. A part of this project will use such algebraic tools to address the classification of large-scale 4-dimensional shapes. The relevant algebraic invariants (in particular, the Milnor group) precisely encode the topological information about the motion of strings in 3-space. Another part of this project will explore interdisciplinary connections between Topology and Statistical Mechanics, with expected applications in both fields.
