My primary research interests are in topological combinatorics, especially applying techniques from discrete Morse theory to detect various properties about topological spaces generated by certain combinatorial structures. In particular, I am interested in calculating connectivity bounds, the Euler characteristic, dimensions of non-vanishing homology, and relevant cellular counting recursions for the spaces in question.
I was first introduced to discrete Morse theory during a Research Experience for Undergraduates at James Madison University (Harrisonburg, VA) in Summer 2011, where my research group worked primarily on the poset topology of pattern-avoiding permutation groups under the strong Bruhat order. Recently, I have been studying the homomorphism complexes generated by mapping chain posets into the Boolean algebras and extending these results to more general distributive lattices. Previously, I also worked with matching trees and the independence and matching complexes of small grid graphs along the lines of Bousquet-Mélou, Linusson, and Nevo.
Publications:
“Homomorphism complexes and maximal chains in graded posets” (with B. Braun). European Journal of Combinatorics 18: 178-194 (2019) Pre-print.
“Matching and independence complexes related to small grids” (with B. Braun). Electronic Journal of Combinatorics 24 (4), P4.18 (2017) Pre-print.
“Permutation pattern avoidance and the Catalan triangle” (with D. DeSantis, R. Field, B. Jones, R. Meissen, and J. Ziefle). Missouri Journal of Mathematical Sciences 25 (1), 2013, 50-60. Pre-print.