Links to my most recent conference proposals are toward the bottom of this page.
My general research interest is combinatorics, an enormous field with any number of subareas that would make interesting thesis projects. The main reason that I like combinatorics is that combinatorial proofs are often very clever. An involution is applied and all the tricky parts fall away, or a bijection is found between seemingly disparate collections. Combinatorics also often allows you to take a very (and I mean VERY) abstract mathematical concept and find a simple and pictorial way to deal with it.
As an undergraduate, I focused on graph theory (though not the probabilistic kind that is UCSD's specialty) and coauthored a paper with Daniel Kleitman of MIT about block transitive tournaments. It was an excellent experience; I didn't even know I wanted to go to graduate school until worked on that.
My undergraduate paper (sort of like an undergraduate thesis) was about four variations on the problem of counting penny fountains in the plane with various restrictions on how they are formed. It was published in the MIT Undergraduate Journal of Mathematics, but it is not available online.
I am working on my Ph.D. thesis with Prof. Jeff Remmel . The project I worked on in my second and third years is in the area of symmetric functions. The ribbon (or zigzag) schur functions form a basis for the space of symmetric functions. These ribbon functions have a dual basis, and we looked for interesting properties of this dual basis. We found a combinatorial interpretation for the coefficients of the expansion of them in terms of the monomial symmetric functions. We also found closed forms for these coefficients in many cases, i.e. small partitions' coefficients, or coefficients with small parts. For example, we have characterized the coefficients indexed by any partition made of 1's and 2's in the first index, and the discrete partition (1^n) in the second index. We have also made progress on the coefficients indexed by (A,B,C,D,..) and (n) given some quite strict restrictions on A,B,C,D. . .
I have another project now which is the bulk my thesis project. It is still focused on the ribbon schur functions, however now I am extending work on the Brenti/Remmel/Mendes homomorphism (whose properties are exploited using the combinatorics of brick tabloids) to be applied to the zigzag schur functions. The progress on this project allows us to find the generating function for the number of permutations containing a specified descent set (where the specified descent set determines which zigzag schur functions we use).
NEWEST papers!
During fall 2006, I worked on a paper that I submitted to the Formal Power Series and Algebraic Combinatorics conference (FPSAC) in Tianjin China. My paper was accepted for a poster presentation, and I will be attending with the support of the National Science Foundation and my department. Those interested can see the extended abstract FPSAC 07 abstract. Also I'm applying to present my research at the Pattern Avoiding Permutations conference in St. Andrews, Scotland in June 2007. A brief abstract is available PermPat 07 abstract .
On October 29th 2006, I will be presenting a talk at the Food for Thought seminar, which introduces first and second year graduate students to potential research areas.
In June 2006, I presented a poster on my first research project at the FPSAC 2006. I've put an extended abstract (very extended), which if you are interested can be found here .
In Spring 2006, I advanced to candidacy for my Ph.D. My thesis committee is: Jeffrey Remmel, Adriano Garsia, Allen Knutson, Fan Chung Graham, Ron Graham, and Alin Deutsch. I appreciate very much their time and effort.