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Joint Meetings:
I will be attending the 2009 Joint Meetings in Washington, DC. Please contact me if you would like to arrange an interview.

Research

The Short Story:

My research concerns distinguishing Legendrian links and surfaces using invariants derived from generating families and the Chekanov-Eliashberg Differential Graded Algebra. The very long story can be found in my Research Statement (.pdf).

2000 Mathematics Subject Classification:

57M Low-dimensional topology
      57M25 Knots and links in S^3
      57M27 Invariants of knots and 3-manifolds
      57M50 Geometric structures on low-dimensional manifolds
57R17 Symplectic and contact topology
53D10 Contact manifolds, general

Keywords:

Low-dimensional topology, contact topology, symplectic topology, knot theory, Legendrian knot theory, Legendrian submanifold invariants, contact homology, Chekanov-Eliashberg DGA, augmentations, generating families, rulings, Khovanov homology, knot Floer homology.

The Long Story:

In 2006 I attended the IAS/PCMI Summer School on Low-Dimensional Topology in Park City, UT. Topics included Dehn fillings, Khovanov homology, Heegaard Floer homology, contact topology, hyperbolic manifold theory and 4-dim topology. The conference was an excellent opportunity to meet a variety of people and learn math from leading researchers. The dynamic of a few hundred mathematicians all in one place working on a common area for three weeks is indescribable. I made important personal connections and learned beautiful and inspiring mathematics.

I spent the 2006-2007 academic year visiting the University of Texas at Austin. There I delved deeply into contact topology. I worked through a variety of papers including Prof John Etnyre's many indispensable expository papers and a set of course notes by Prof Bob Gompf. Prof John Luecke and I spent the spring term reading through papers on bounds on the Thurston-Bennequin number of Legendrian knots. While at UT, I was not yet focusing on Legendrian knot theory, but instead studying contact topology as a whole. This included studying convex surface theory, open book decompositions, the classification of tight structures on lens spaces and other nice closed 3-manifolds, and legendrian submanifolds.

During Thurston's 2007 Birthday Conference at Princeton, I was fortunate to meet Prof Joshua Sabloff from Haverford College and he guided me towards studying the Chekanov-Eliashberg DGA. Josh has been an invaluable resource as I've worked to understand the field. This past summer he joined my thesis committee as co-advisor and we now meet weekly to discuss research. Since that conference I have concentrated on Legendrian submanifold invariants, especially those related to the generating families and the Chekanov-Eliashberg DGA, both in dimension 3 and higher.

In September 2008, I was invited to attend the research workshop on Legendrian and Transverse Knots at the American Institute of Mathematics (AIM) in Palo Alto, CA. There I spoke on my work to extend the graded normal ruling invariant from Legendrian links to Legendrian surfaces. During the workshop, the working group I was in made considerable progress towards refining the graded normal ruling invariant. In fact, the working group has applied for a SQuaREs grant from AIM that will allow us to reconvene at AIM for up to two weeks to continue our collaboration on this project.

I also keep an eye on the areas of Khovanov homology and knot Floer theory. I find both of these areas fascinating in their own right. It just happens that they also produce interesting invariants of Legendrian and transverse links.