Elizabeth Kelley, Ph.D.
Assistant Professor
- Milwaukee WI UNITED STATES
Dr. Kelley works in algebraic combinatorics, with a particular interest in cluster algebras and total positivity.
Multimedia
Education, Licensure and Certification
Ph.D.
Mathematics
University of Minnesota
2021
M.S.
Mathematics
Harvey Mudd College
2018
B.S.
Mathematics
Harvey Mudd College
2015
Biography
Her work examines combinatorial and algebraic structures including generalized cluster algebras, graph LP algebras, cluster scattering diagrams, theta functions and combinatorial models associated with orbifolds and punctured surfaces. Her research has appeared in journals and conference proceedings in algebraic and enumerative combinatorics.
Kelley previously served as a postdoctoral research associate and lecturer at the University of Oklahoma and as a Combinatorics RTG postdoctoral fellow at the University of Illinois Urbana-Champaign. She has also contributed to mathematical seminars, conferences and undergraduate research mentoring.
She earned a Ph.D. in mathematics from the University of Minnesota in 2021, where she worked under the supervision of Gregg Musiker. She earned a Bachelor of Science in mathematics, with a minor in Arabic, from Harvey Mudd College.
Areas of Expertise
Accomplishments
“Thank a Teacher” Certificate, University of Minnesota
2021
List of Teachers Rated Excellent, University of Illinois Urbana Champaign
2022 and 2023
Social
Selected Publications
Hydra Graphs for Graph LP Algebras
Annals of CombinatoricsE. Kelley, E. Banaian, S. Chepuri, and S.W. Zhang
2025-12-19
Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. We provide a combinatorial proof of positivity for certain cluster variables in these algebras. This proof uses a hypergraph generalization of snake graphs, a class of planar graphs which were used by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. These results extend those given in our previous paper, where we used a related combinatorial object known as a T-path.
An elaborate new proof of Cayley’s formula
Algebraic CombinatoricsEsther Banaian, Anh Trong Nam Hoang, Elizabeth Kelley, Weston Miller, Jason Stack, Carolyn Stephen, Nathan Williams
2024-02-12
We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type and vertex-labeled trees on vertices. By an argument of Galashin, Lam, and Williams using Opdam's trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on vertices.
Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras
Annals of CombinatoricsMan-Wai Cheung, Elizabeth Kelley, Gregg Musiker
2022-12-09
We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.
