Classifying terminal Fano varieties using Mirror Symmetry

In recent years, two fundamentally different approaches to mapping the landscape of terminal Fano threefolds have emerged. On the one hand, there are the graded ring techniques pioneered by Brown and Reid; on the other hand are new ideas emerging from Mirror Symmetry, as developed by Coates, Corti, Kasprzyk, etc. This thesis exploits both approaches, combining their fundamentally different mathematical ideas to begin to understand the classification of terminal Fano threefolds. The main resources are the Graded Ring Database -- a database of candidate Hilbert series of Fano threefolds -- and various classifications of lattice polytopes.