Contact
Office: 204 TMCB
Email: hughes at mathematics dot byu dot edu
Phone: (801) 422-7416
Winter 2025 Office Hours
Mondays 11:00am - 11:50am
Wednesdays 12:00pm - 12:50pm
Fridays 9:00am - 9:50am
Research Interests
My research interests lie mainly in low-dimensional topology , and in particular knot theory and the topology of 4-dimensional smooth manifolds .
Recently, I've been using techniques from machine learning and neural networks to predict and compute difficult knot invariants, like the slice genus of a knot. I am especially interested in using techniques from reinforcement learning to solve difficult topological problems.
Another problem I've been interested in involves understanding how various surfaces can be knotted and "braided" in a given 4-dimensional manifold, and what these surfaces can tell us about the topology of the overall space. The boundaries of these surfaces are knots (closed braids, actually) in 3-space, and I've been trying to find ways to use the algebraic information from the boundary link to provide geometric information about the surface itself. Some of the tools I've been using include Khovanov and Khovanov-Rozansky homology, Dehornoy orderings of the braid group, and Garside normal forms.