The page showcases the work of a few undergraduate students I have supervised. Both the original content and the writing of papers on this page are entirely due to the authors. The projects I proposed were rather sophisticated for undergraduate work. I have been impressed at how they have risen to the challenge.

Columbia REU, 2007.

In the summer of 2007, Columbia ran an internal "research experience for (Columbia) undergraduates." I had the pleasure of advising one of the two projects. I was assisted by Thomas Peters; without him, the projects would probably not have been a success. (The other project that summer was advised by Mirela Ciperiani and Kimball Martin.)

Columbia REU, 2009.

The REU has continued since 2007. In 2009, I was fortunate to co-advise one of the projects with Timothy Perutz, with help from Jonathan Bloom and Thomas Peters. The students worked hard on their paper long after the REU ended.

Edward Trefts's senior thesis.

A continuation of his work from the summer, giving a more in depth account of the Alexander polynomial and its relationship to the genus. He also gives the first publicly available, elementary proof that a new definition of the Alexander polynomial, via grid diagrams, agrees with the classical one.

Emily Clader's senior thesis.

M. C. McCord showed in 1965 that any finite simplicial complex is modeled, up to weak homotopy equivalence, by a space with finitely many points. In her thesis, after explaining McCord's proof, Clader strengthens his result, showing that any finite simplicial complex is actually a deformation retract of an inverse limit of finite spaces.

Atanas Atanasov's senior thesis.

W. Thurston proved that if K is a hyperbolic knot then the manifolds obtained by increasingly large surgeries on K converge (in the Gromov-Hausdorff sense) to the complement of K. In his thesis, after reviewing the substantail background material, Atanasov begins a program to understand how this kind of convergence interacts with a particular class of modern 3-manifold invariants, Heegaard Floer homology.

Kyler Siegel's senior thesis.

Bordered Heegaard Floer homology gives an action of the strongly based mapping class group of an orientable surface on a derived category of modules. This action was described combinatorially in arXiv:1012.1032. In his thesis, Siegel gives a combinatorial proof that the action is well-defined.

  • Kyler Siegel, "A Geometric Proof of a Faithful Linear-Categorical Surface Mapping Class Group Action." arXiv:1108.3676. August 2011.

Knot Theory Course, Spring 2011.

Instead of a final exam, students in my undergraduate knot theory class in the spring of 2011 wrote expository final papers on topics we hadn't covered. I was impressed with the results; here's a sample:

Nathaniel Schieber's senior thesis.

A toral tangle is a tangle embedded in the solid torus. Krebes investigated whether a particular toral tangle can be completed to an unknot in the three-sphere (with the standard embedding of the solid torus). This was studied further by Abernathy and Abernathy-Gilmer. In his thesis, Schieber discusses discusses computing the Alexander polynomial of knots and the Jones polynomial of tangles, and applies these to obtain computational evidence that Krebes's tangle can not e completed to an unknot. (I heard about this question from Gilmer.)