Department of Mathematics & Statistics, University of Vermont
My mathematical research focuses on combinatorial questions motivated by algebra and geometry. I am invariably trying to understand patterns arising in the relationships among combinatorial objects. Just as in nature, there are many structures that hint at hidden depths.
The Kazhdan-Lusztig polynomials are a particularly rich family of polynomials arising from representation theory and the geometry of the flag manifold that I am attempting to understand combinatorially. This image depicts the reciprocals of the roots of the 726,636 distinct Kazhdan-Lusztig polynomials associated to the exceptional Coxeter group H_4. The positive x-axis is pointing directly up. Shading is derived from the density of roots.
Each of the 25 squares is a rotationally symmetric labyrinth on a 65x65 grid. Each path starts at the upper left and finishes at the lower right. The labyrinths were created by taking random walks in the space of all such labyrinths. These were inspired by an idea of Chuck Hulse.