On the 11th of January 2024, I defended my MSc thesis in Invariant Theory and Graph Theory. The thesis can be found here or you can read the abstract below.

Abstract

In this thesis, we provide an exposition of the invariant theory of finite groups, with a focus on algorithms and the Hilbert series. We apply the built-up theory to the algebra of invariants of multigraphs, as well as \(s\)-graphs, which are graphs weighted in \(\{0,1,…,s\}\). Utilizing computer exploration on the invariant algebra of \(s\)-graphs, we derive a formula for the Hilbert series of any permutation group acting on a special discrete variety, \(V|_s\). We conjecture that this formula can be generalized to any finite group. Furthermore, we present a version of King’s algorithm for computing a (minimal) generating set for the algebra of invariants on simple graphs. We conjecture the correctness of this algorithm and its potential generalization to any finite group acting on \(V|_s\). Finally, we recreate Thiery’s disproof of Pouzet’s conjecture