The paper A finite generating set for the genus g (p,q,n)-dipole series from perturbative Yang-Mills theory, written by David Jackson and me, has recently appeared on the arXiv. The approach to the (p,q,n)-dipole problem appearing in this paper begins with a combinatorial argument: adding an edge to a (p,q,n)-dipole either cuts a face of the dipole in two, or joins two faces and requires the addition of a handle to the surface in which the dipole is embedded. These combinatorial operations are interpreted as differential operators on the generating series for the problem, leading to a pair of partial differential equations which determine the solution to the problem. An analysis of this equation leads to a recursive procedure for finding the solution to the problem for a surface of genus g, and identifies a natural set of functions in which to express these solutions. The paper is available at http://arxiv.org/abs/1110.2806.
Two papers on near-central enumerative techniques now available
Today two papers, jointly authored by David M. Jackson and me, are now available on the arXiv. They describe an algebraic approach to a class of problems which we refer to as near-central problems. Although the two papers are ideally read together, each has been written so that it may be read and understood without reading the other.
The first, Character-theoretic techniques for near-central enumerative problems, develops the basics of the theory of enumeration for near-central problems. We demonstrate this theory by applying it to a combinatorial problem called the (non-transitive) star factorization problem. This paper is available at http://arxiv.org/abs/1108.4045 .
The second, Near-central permutation factorization and Strahov’s generalized Murnaghan-Nakayama rule, uses a theorem of Strahov to further develop these techniques. The techniques are then applied to a special case of the (p,q,n)-dipole problem — the case when q=n-1 — to give a solution for all orientable surfaces. Moreover, this solution reveals an unexpected symmetry in this class of dipoles for which, at present, no combinatorial proof is known. This paper is available at http://arxiv.org/abs/1108.4047 .
Future Directions in Combinatorial Centralizer Theory
I’ll be giving an informal talk this Tuesday (July 12) about some open combinatorial problems that have arisen out of the research appearing in my thesis. The talk starts at 1:30 p.m. in MC 5136. Here is the full title and abstract:
Future directions in the combinatorics of centralizers of the symmetric group algebra
Enumerative questions about factorizations of permutations which depend on information other than the cycle type of the permutations are called non-central problems. Examples of such problems include the (p,q,n)-dipole problem and the star factorization problem, which is also known as the problem of determining powers of Jucys-Murphy elements. Centralizers of the symmetric group algebra with respect to a subgroup H provide an algebraic context for studying such problems. When H is the subgroup consisting of permutations fixing n, the corresponding centralizer Z_1(n) is well-understood. It has been applied to several combinatorial problems, and in particular it gives a solution to a special case of the (p,q,n)-dipole problem for all orientable surfaces. The recent work on Z_1(n) has led to several open combinatorial problems, the study of which may help advance the theory of centralizer algebras. In this talk, I will introduce these problems and explain why I think they will be interesting to study.
It’s now official
Last Friday I crossed the stage at UW’s convocation ceremony, so I am now officially a PhD graduate. Consequently, there is now a photo of me on the “About” page wearing UW’s doctoral regalia. I’ve also uploaded my most recent curriculum vitae to the site.
Welcome to my new website
I have recently overhauled my website, and I am gradually adding new content. In the mathematics section, you can find copies of my doctoral and master’s theses, as well as the slides from two contributed talks I gave in May, at the Ontario Combinatorics Workshop and the Ottawa-Carleton Discrete Mathematics Days.