Matt Davis Research
Department of Mathematics
Harvey Mudd College
Generally speaking, I do combinatorial representation theory, meaning I like to turn weird-looking algebra into pretty pictures. Check here for a good survey of the field. My thesis was on the affine Hecke algebra, examining its representations at roots of unity in an attempt to understand what in the world is going on with them. Of late, I'm being drawn into the world of voting theory and applied representation theory . Other things I would like to know about include perverse sheaves, particularly as they relate to the affine Hecke algebra, among other aspects of their uses in representation theory.
Representations of Rank Two Affine Hecke Algebras, preprint. Submitted.
Summer 2011: Catalan numbers - symmetry and generalizations. Jack's project revolved around symmetries of sets of objects counted by the Catalan numbers. The motivating question was finding a combinatorial symmetry that exchanges two combinatorial statistics. Although the description of this bijection remains elusive, Jack thoroughly studied a number of interesting symmetries of Catalan objects. He formulated a framework for comparing bijections between different sets, and found some striking similarities between very different-looking symmetries. Poster
Palmer Mebane: Palmer has been considering the m-Catalan numbers and m-parking functions. The standard Catalan numbers and parking functions count the chambers in a hyperplane arrangement called the Shi arrangment, and certain combinatorial statistics can be interpreted on both sides of this bijection. There is a natural generalizations of the Shi arrangement called the m-Shi arrangement. Palmer has been exploring (and sometimes discovering) the resulting m-versions of the original combinatorial statistics and their generating functions.