Author ORCID Identifier
Year of Publication
Doctor of Philosophy (PhD)
Arts and Sciences
Dr. Christopher Manon
Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for T-varieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the well-poised property. In particular, I show that the well-poised property is preserved under a geometric invariant theory quotient by a (quasi-)torus. Conversely, I argue that T-varieties built on a well-poised base preserve the well-poised property when the base satisfies certain degree conditions.
The second half of this dissertation covers two projects in algebraic statistics. The first studies level-1 phylogenetic network models which model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. In particular, I found the quadratic invariants of the Cavendar-Farris-Neyman model for level-1 networks and conjecture these generate the corresponding ideal. In the final project, I study a class of statistical models known as binary hierarchical models. Hierarchical models are known to be log-linear; thus, the joint probability distributions of the random variables naturally lie on a toric variety. For many applications such as testing normality of the model and finding a maximum likelihood estimate, a H-description of the marginal polytope is needed to drastically speed up computations. Here I provide an alternative polytope isomorphic to the marginal polytope in the binary case. This polytope is known as the generalized cut polytope, and I compute H-descriptions for all binary hierarchical models whose underlying simplicial complex is pure and of codimension 1.
Digital Object Identifier (DOI)
Cummings, Joseph, "Tropical Geometry of T-Varieties with Applications to Algebraic Statistics" (2022). Theses and Dissertations--Mathematics. 91.