Extensions of Modern Homological Invariants in Low Dimensional Topology

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Topology is the branch of mathematics that studies shapes of spaces. Topology has several real-world applications, such as studying DNA knotting, constructing new data encryption algorithms, analyzing large data sets, motion planning for robotics, and developing quantum field theories in physics, to name a few. Due to a famous theorem by Smale, topology in higher dimensions is somewhat simpler than topology in lower dimensions. Therefore, it is interesting to concentrate only on spaces up to dimension four; this sub-field is called low-dimensional topology. These low-dimensions also correspond to the spaces that we live in---we live in a three-dimensional space, and if one includes time, in a four-dimensional spacetime---making low-dimensional topology even more pertinent. In low-dimensional topology, we are specifically interested in knot theory, where one studies one-dimensional objects inside three-dimensional spaces, such as knotted pieces of strings. Knot theory is a fundamentally important topic in low-dimensional topology, and it is also an integral part of many of the real-world topological applications. Knot theory studies whether a knot can be transformed into another without tearing or crossing itself (such a transformation is called an isotopy), and if not, what sort of modifications need to be made to ensure they become isotopic. Knot invariants are mathematical objects (such as numbers or groups) associated to knots which remain unchanged during such an isotopy, and consequently, are extensively used in studying knots. The current project is focused on knot theory and will explore existing knot invariants and construct new ones. This project will concentrate on two modern families of knot invariants in low-dimensional topology, knot Floer homology and Khovanov homology, which have been employed for a variety of applications ever since their discovery at the turn of the millennium. The main aim of the project is to construct new extensions, such as spatial refinements, of various versions of these existing invariants. Specifically, the project has the following four goals: construct further spatial refinements of Khovanov homology invariants and their perturbations; construct a spatial refinement of knot Floer homology using grid presentations; study group actions on Lagrangian Floer homology; and construct new combinatorial spectral sequences from Khovanov homology. Additionally, several activities combining research with educational and other broader impacts will be organized as part of this project, such as increasing mathematical awareness and interest among children at Los Angeles Math Circle. 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.