I am interested in geometry, topology, dynamics, and Lie groups. These days, I am mostly thinking about knot theory and low-dimensional topology.

Once, I worked on a project in mathematical biology about the very early embryonic development of the fruit fly drosophila melanogaster. That was really a differential equations modeling project in the end.

I really like getting students involved in research, and that influences the kinds of things I can spend time on. Some questions just have too steep a learning curve for an undergraduate to do anything meaningful. (Every question related to my PhD thesis fits in this category.) But knot theory is a very good field for undergraduates to work on. It has:

  • easy realizations of the basic objects, since you can just use string,
  • lots of visual appeal,
  • interesting questions that can be understood quickly,
  • connections with other things that one can learn quickly (like graph theory),
  • stuff you can do immediately to get started, and
  • plenty of depth.

Anyway, these days I am exploring some “classical” knot theory topics from a new perspective. For experts, the relevant words here are: bridge indices, branched coverings, rational tangles, and butterfly diagrams.