Dissertation

My dissertation is titled "Optimization Techniques in Extremal Graph Theory". You can find it here.  You can find slides used for my defense here. 

Abstract: This thesis focuses on the mathematics and uses of the plain flag algebra method. We begin by outlining the basic definitions and theorems needed to be able to add, multiply, and average flags. We then discuss how to translate this into the plain flag algebra method and how one can then implement it computationally. Next, we talk about the types of problems that the plain flag algebra method can solve, including a new approach for proving the non-existence of certain graphs. We also consider two novel modifications of the plain flag algebra method. The first of these replaces semi-definite programming in the original method with copostive programming, which is computationally much more difficult and does not appear to provide much better bounds. The second modification reduces the size of the semi-definite program used in the plain flag algebra method, which greatly speeds up the programs with only a minor decrease in accuracy. Finally, we move on to applications of the plain flag algebra method. After introducing the ideas of inducibility and fractalization we prove three results using the plain flag algebra method: C_4 fractalizes, C_5 almost fractalizes, and directed P_{1,2,1}’s inducibility.