Kathryn Mulholland

Assistant Professor of the Practice

Education

Ph.D., University of Notre Dame, 2020
M.S., University of Notre Dame, 2016
B.S., California Polytechnic State University, San Luis Obispo, CA, 2014

Research Group

Algebra and Algebraic Geometry

Research Area

Algebra and the Scholarship of Teaching and Learning

Bio

Research Interests

My graduate research connected the two fields of Cluster Algebras and Poisson Geometry. More specifically, in my dissertation I constructed a generalized cluster algebra from a non-standard Poisson structure on the space of complex rectangular matrices. A cluster algebra is a constructively defined commutative ring with a set of generators, cluster variables, divided into overlapping subsets of the same cardinality, clusters.  A cluster algebra is said to be compatible with a Poisson structure if the bracket of two cluster variables is given by the simplest kind of quadratic bracket. Constructing a compatible cluster algebra from a non-standard Poisson structure is a widely unsolved problem in mathematics.
 

Cluster algebras were introduced by Fomin and Zelevinsky in 2003 in an attempt to create an algebraic framework for the study of dual canonical bases and total positivity in semisimple groups. Since its inception, however, cluster algebras have appeared in many other contexts, including Grassmanians, quiver representations, canonical bases, generalized associahedra, Teichmuller theory, spectral networks, 3d gauge theories, and Poisson geometry.