Talks and presentations

November 19, 2024

Talk, JHU Math Club, Baltimore, MD

Abstract: Integer matrices \(A\) and \(B\) are said to be \(\mathbb{Z}\)-similar if there exists \(C \in \text{GL}_n(\mathbb{Z})\) such that \(CAC^{-1} = B\). While the canonical forms of linear algebra provide a complete description of matrix similarity over a field, much is still unknown about \(\mathbb{Z}\)-similarity. We will briefly review finite-dimensional linear algebra and see how this theory breaks down when we try to apply it to \(\mathbb{Z}\)-similarity. We will then introduce a modern form of the Latimer-MacDuffee theorem to convert the \(\mathbb{Z}\)-similarity problem to computing isomorphism classes of certain modules. We will explain how the simplest case of this isomorphism problem reduces to the number-theoretic problem of computing ideal class groups. We will conclude by characterizing the integer matrices whose \(\mathbb{Q}\)-similarity class splits into finitely many \(\mathbb{Z}\)-similarity classes.

April 08, 2024

Talk, JHU Math Club, Baltimore, MD

Abstract: With a proof spanning thousands of pages across hundreds of articles, the classification of finite simple groups is a monumental achievement of 20th century mathematics. We will explore how finite simple groups act as the building blocks for all finite groups, and discuss the general strategy of the classification of such simple groups. We will introduce Feit and Thompson’s odd order solvability theorem and show how this theorem along with the Brauer-Fowler theorem provides a possible strategy of attack on the classification theorem. Recording