Toric geometry of the 3-Kimura model for any tree

In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated

Constructive degree bounds for group-based models

Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant. We prove that for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant $d$, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were extended to G-models (including 2-Kimura and all JC

Classification of normal phylogenetic varieties for tripods

We provide a complete classification of normal phylogenetic varieties coming from tripods, and more generally, from trivalent trees. Let $G$ be an abelian group. We prove that the group-based phylogenetic variety $X_{G,\mathcal{T}}$, for any trivalent tree $\mathcal{T}$, is projectively normal if and only if $G\in \{\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_5, \mathbb{Z}_7\}$

Phylogenetic degrees for claw trees

Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, respectively the study of mutations of organisms. Both theoretically and in terms of applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety $X_{G, n}$ with $G\in\{\mathbb{Z}_2,\mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{Z}_3\}$ and any $n$-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes $P_{G,n}$. We apply combinatorial methods and we give concrete formulas for them

2014 Conference Abstracts: Annual Undergraduate Research Conference at the Interface of Biology and Mathematics

Conference schedule and abstract book for the Sixth Annual Undergraduate Research Conference at the Interface of Biology and Mathematics Date: November 1-2, 2014Plenary Speakers: Joseph Tien, Associate Professor of Mathematics at The Ohio State University; and Jeremy Smith, Governor\u27s Chair at the University of Tennessee and Director of the University of Tennessee/Oak Ridge National Lab Center for Molecular Biophysic