PSU students at the 2011 Hudson River Undergraduate Conference
The combinatorial nature of my research naturally lends itself to collaborations with undergraduates, and my goal is to incorporate students in my research as much as possible. I plan to recruit 1-3 students per academic year to investigate an original research project. If you are a PSU student interested in conducting research in mathematics, please come talk to me! Occasionally, there may be funding available to pay students to do research. Here is a list of my recent undergraduate research projects.
Fall 2011-Spring 2012
Exploration of T-avoiding elements in Coxeter groups of type $F$: Currently mentoring Ryan Cross, Katie Hills-Kimball, and Christie Quaranta on an original research project aimed at exploring the "T-avoiding" elements in Coxeter groups of type $F$. Students will present findings during at least one conference.
Fall 2010-Spring 2011
T-avoiding permutations in Coxeter groups of types $A$ and $B$: I mentored Joseph Cormier, Zachariah Goldenberg, Jessica Kelly, and Christopher Malbon on an original research project that classified of the "T-avoiding" permutations in Coxeter groups of types $A$ and $B$. The students made the following presentations:
- Classification of the T-avoiding permutations and generalizations to other Coxeter groups, Undergraduate Student Poster Session, Joint Mathematics Meetings 2012, Boston, MA; January 6, 2012, Abstract (PDF), Poster (coming soon)
- Classification of the T-avoiding permutations and generalizations to other Coxeter groups, Combinatorics of Coxeter Groups Special Session, Spring 2011 Eastern Sectional Meeting of the AMS, College of the Holy Cross, Worcester, MA; April 10, 2011, Abstract (PDF), Slides (PDF)
- Classification of the T-avoiding permutations, 2011 Hudson River Undergraduate Mathematics Conference, Skidmore College, Saratoga Springs, NY; April 16, 2011, Abstract (PDF), Slides (PDF)
We are currently in the progress of writing up our results with the intention of submitting an article for publication.
Spring 2010
Counting generators in Temperley--Lieb diagrams of types $A$ and $B$:
Sarah Otis and Leal Rivanis obtained original results concerning Temperley--Lieb diagram algebras of types $A$ and $B$, which have a basis indexed by the fully commutative elements in Coxeter groups of types $A$ and $B$, respectively. We also obtained preliminary results in the type affine $C$ case. The students presented their work at the following conference:
My primary research interests are in the interplay between combinatorics and algebraic structures. More specifically, I study the combinatorics of Coxeter groups and their associated Hecke algebras, Kazhdan-Lusztig theory, generalized Temperley--Lieb algebras, diagram algebras, and heaps of pieces. By employing combinatorial tools such as diagram algebras and heaps of pieces, one can gain insight into algebraic structures associated to Coxeter groups, and, conversely, the corresponding structure theory can often lead to surprising combinatorial results. You can read more about my current research interests here.