DescriptionTopologists are interested in the ways in which one space can live inside another space. The simplest case in which there are interesting examples is that of knot theory. This studies the different ways in which a circle can be embedded in 3-dimensional space. Knot theory has undergone a couple of upheavals. Firstly, Vaughan Jones earnt himself a Fields medal went he discovered the Jones polynomial - an invariant that turned out to be integral to the proof of a couple of long-standing conjectures. Then, in the 21st century, Khovanov moved things further with his invention of Khovanov homology and its relatives which categorified the Jones polynomial. This pointed to deep connections with other areas of mathematics - Floer homology, representation theory, physics. Khovanov's construction is beautiful, simple, and has revolutionized the field. In this project we will begin by getting a good grounding in the basics of Khovanov homology - understanding the construction, the gradings, and a few computional techniques. After this, the world is your oyster! There's Rasmussen's proof of the Milnor conjecture, Bar-Natan's elegant tangle reformulation, relationships with Floer homology, categorifications of other knot polynomials, computations for classes of knots, experimenting with coefficient rings, extensions to braids, the list goes on... Note: I shall be travelling away from Durham for much of Epiphany term. During these weeks the supervisions will be conducted online.Mode of Operation and Evidence of LearningThe project will revolve around learning through reading with focus on the underlying theory and the development of conceptual understanding. Students will demonstrate their understanding by exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.PrerequisitesGeometric Topology III.Co-requisitesAlgebraic Topology IV.ResourcesCheck out the overview written for the American Mathematical Society by the project leader. There are also two expository papers by Paul Turner accessible on the arXiv here and here. |
email: Andrew Lobb.