#### Author ORCID Identifier

#### Year of Publication

2022

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department/School/Program

Mathematics

#### First Advisor

Dr. Christopher Manon

#### Abstract

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)

https://doi.org/10.13023/etd.2022.185

#### Recommended Citation

Cummings, Joseph, "Tropical Geometry of T-Varieties with Applications to Algebraic Statistics" (2022). *Theses and Dissertations--Mathematics*. 91.

https://uknowledge.uky.edu/math_etds/91