Lattice Polytopes from Graphs and Networks

This proposal is to investigate algebraic and geometric properties of lattice polytopes arising from graphs and networks. There are three main project topics. The first is flow polytopes of graphs, for which I will be investigating volumes, triangulations, and Ehrhart theory. The second is graphical Hermite normal form simplies, for which I will be investigating Ehrhart theory, triangulations, and Hilbert bases. The third is symmetric edge polytopes, for which I will be studying facet enumeration and the relationship between facet numbers and clustering metrics. The broader impacts of this work will include math education research projects, collaborations with historians, doctoral mentoring and human resources development, and professional development regarding teaching and mentoring.