Associate
Professor

Office: 204 TMCB

E-mail: hughes at mathematics dot byu dot eduPhone: (801) 422-7416

Mondays 10:00am-10:50am

Wednesdays 2:00pm-2:50pm

Fridays 9:00am-9:50am

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 try 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 to do this include Khovanov and Khovanov-Rozansky homology, Dehornoy orderings of the braid group, Garside normal forms, etc.

Papers and Preprints

Education

CV

Implementing Technology in the Linear Algebra Classroom

A Primer on Proof Writing