Polyhedral and tropical geometry in nonlinear algebra

Dissertation Defense
Wednesday, June 30, 2021 - 3:00pm for 1 hour (actually 50 minutes)
Cvetelina Hill – Georgia Tech – cvetelina.hill@gatech.edu
Cvetelina Hill

Please Note: BlueJeans link: https://bluejeans.com/298474885/8484

This dissertation consists of various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. We give a broad introduction and discuss connections to applied algebraic geometry, polyhedral, and tropical geometry. The individual topics discussed are as follows:

  • Interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. Amongst other results, we characterize tropically convex sets in any dimension, and give a combinatorial description for the dimension of the tropical convex hull of an ordinary affine space. 
  • The steady-state degree and mixed volume of a chemical reaction network. We present three case studies of infinite families of networks. For each family, we give a formula for the steady-state degree and mixed volume of the corresponding polynomial system. 
  • Methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. 

Thesis may be viewed here.

BlueJeans link