My primary research interest lies in the areas of
algebraic combinatorics, representation theory, and symmetric function
theory, and in
particular symmetric and quasisymmetric functions. As a more
specific indication of my interests, I currently am actively thinking
about particular problems related to: Macdonald polynomials and the n!
conjecture, k-schur functions, rational Catalan combinatorics, random
generation of combinatorial objects, and the sand pile model.
doctoral thesis, completed under Adriano Garsia, concentrated on
parking functions and their conjectured (proven?--see the latest
exciting reasearch by Carlsson
) relation to the diagonal harmonics.
One really nice and unusual use of representation theory is in
computing time to uniformity in group walks, as kindly explained to me
by my postdoc mentor, Persi Diaconis; we worked on a classical random
walk on the Heisenberg group
using this Fourier Analytic approach and this explains several recent
papers with a less obviously combinatorial bent.
- Applying Representation Theory to Random Walks.
CAGE Seminar at U Penn. September, 2016
- A Random Walk on the Heisenberg Group. U.C. Davis
Combinatorics Seminar. May, 2016
Formidable symmetries from the $q, t$ Catalan and Beyond. MAA Golden
Section Invited Speaker, Foothill University, February 2015.
- A simpler
formula for the number
of diagonal inversions of an ((m,n))-Parking Functions. AMS Special
Session, Dalhousie University, October 2014.
(Surprise!)-- The Diagonal Harmonics} York University, March 2014.
- The Diagonal
Harmonics and $n$
Capricious Wives} University of San Francisco Mathematics Colloquium
Functions, and Parking Polynomials} Bay Area Discrete Math Day October
Diagonal Harmonics, and a Surprising (!) Connection} UC Berkeley
Combinatorics Seminar October 2013.
between a family of recursive polynomials and parking function theory,
International Conference on Formal Power Series and Algebraic
Combinatorics, Nagoya, Japan, August 2012.
- New Approaches to
the Study of Parking Functions and the Theory of the Diagonal
Harmonics, University of Washington Combinatorics Seminar, May 2012.
Family of Polynomials Suggested by the Haglund- Morse-Zabrocki
Conjecture, Workshop de Algebra V Teoria de Numeros, Chile, December
- A New Parking Function Statistic, Special Session on
Symmetric Functions, Symmetric Group Characters, and Their
Generalizations, AMS Sectional Meeting, Wake Forest University,
- A Parking Function Bijection Suggested by the
Haglund-Morse-Zabrocki Conjecture, Banff International Research
Institute for Mathematical Innovation and Discovery, November 2010.
- Two Parking Function Bijections, Universite de Bordeaux,
- Combinatorics of the Diagonal Harmonics, MIT Women in
Lecture Series, March 2009.
Metric Dimension of the Cayley Digraphs of Finite Abelian Groups, AMS
Session on Combinatorics, Joint Meetings, January 2007.
- Applications of Lie Symmetry Groups to Minimal Surfaces, Pi
Epsilon Session, MAA Mathfest, August 2005.