Casey M. Pinckney
Research
My research interests broadly are in algebraic and topological combinatorics. I am interested in particular in simplicial complexes, polytopes, and groups.
Independence Complexes of Finite Groups
For my PhD thesis, I studied simplicial complexes that arise from minimal generating sets of finite groups and their subgroups. I call these independence complexes. One of my points of focus was to describe the algebraic and combinatorial structure of these complexes. As an example, cyclic groups, which in many settings are among the "easiest" groups to study, already give rise to independence complexes with relatively sophisticated structure and complicated, yet beautiful, combinatorics.
My PhD advisors: Alexander Hulpke and Chris Peterson.
Invariant Theory of Finite Groups
The elementary symmetric polynomials generate a ring which is invariant under the action of the symmetric group. For my expository Master's thesis, I studied known techniques to compute the invariant subring of a polynomial ring under other group actions, and wrote code in Macaulay2 to explicitely compute these invariant rings.
If you are interested in learning some introductory material on this topic, perhaps this document will be useful. It supplements some of the introductory material and examples found in Bernd Sturmfels' Algorithms in Invariant Theory. The invariantRing package in Macaulay2 also has direct methods to compute invariant subrings.
h-vectors of PS ear-decomposable graphs
As an undergraduate at Seattle University, I co-authored a paper solving a specialization of a generalization of a conjecture made by Richard Stanley in 1977 about the combinatorial structure of matroid complexes. This was joint work with my research mentor Steve Klee and students from Seattle University and University of Washington.
Nima Imani, Lee Johnson, Mckenzie Keeling-Garcia, Steven Klee, and Casey Pinckney. “h-vectors of PS ear- decomposable graphs.” Involve: A Journal of Mathematics, Vol. 7, No. 6 (2014). 743-750.
Preprint available here.
I have worked on two projects that were born at the Graduate Research Workshop in Combinatorics (GRWC) at the University of Kansas, Lawrence in Summer 2019. One project has resulted in a published paper (available on arXiv) and the other is ongoing.
Lattice Polytopes arising from Schur and Symmetric Grothendieck Polynomials
GRWC collaboration with Margaret Bayer, Bennet Goeckner, Su Ji Hong, Tyrrell McAllister, McCabe Olsen, Julianne Vega, Martha Yip, and Semin Yoo
Our paper, Lattice polytopes from Schur and symmetric Grothendieck polynomials, is published in the Electronic Journal of Combinatorics, available here and also available on the arXiv.
Whitney Duals of Posets
GRWC collaboration with Margaret Bayer, Josh Carlson, Bennet Goeckner, Joshua Miller, Kyle Murphy, Sung-Yell Song, and Julianne Vega
As an undergraduate, I participated in an NSF-sponsored Research Experience for Undergraduates (REU) at the University of Portland in 2013, studying Automorphism Groups of Compact Riemann Surfaces.
Faculty mentors: Dr. Aaron Wooten and Dr. Valerie Peterson