DRP – Spring 2024

The aim of this presentation is to introduce the elementary aspects of Lie Theory by presenting the mathematical objects it studies through examples and relevant theorems. We will introduce the general linear group, $\mathsf{GL_n(\mathbb{C})}$, in order to define the matrix Lie groups that derive from it, such as $\mathsf{SL_n(\mathbb{C})}$, $\mathsf{SU(n)}$, $\mathsf{GL_n(\mathbb{R})}$, and $\mathsf{SO(n)}$. We will explore some of the topological properties of matrix Lie groups such as their path connectedness. Ultimately, we will define the Lie algebra of a matrix Lie group which will aid our study by giving us a flat insight into a curved object.  

SET is a pattern matching card game with many complex and varying mathematical dimensions. This project looks at affine geometry by examining the structure of SET. It considers how the Axioms of Geometry translate to SET and discusses how to construct a “magic square” which resembles $\mathsf{AG(2, 3)}$. Finally, it uses this knowledge to describe a hyperplane in terms of SET.

This talk will introduce regular languages, a rigorous mathematical framework through which we can study natural languages, which you are reading right now. A weak but important requirement for this framework is being mechanically able to decide whether a string of letters is part of our given language or not – and receive the answer “Yes” from a machine if it is. For this talk, we will introduce finite state automata, which should be able to “accept” exactly the strings we want, and no others. Ultimately we will address a fundamental result: that the languages accepted by finite automatons are exactly the languages defined by regular expressions.