Algebraic groups and Lie algebras

During my first year at the University of Oxford I was captivated by Linear Algebra, a topic I had rarely encountered before. I was struck by how it seemed to branch out into a plethora of other mathematical topics. Since then, I have thoroughly enjoyed exploring many of these topics, especially Lie algebras and Representation theory. I look forward to applying the theories I have already studied to an in-depth focus under the supervision of Dr Simon Goodwin during this PhD course. Building mathematical concepts from the ground up has always been something that naturally speaks to me. Having taken advantage of the basic laws of addition and multiplication since childhood, it was enlightening to learn about the axiomatic construction of the real numbers during a first-year Analysis class, and how such vast complexity could be distilled to a few simple axioms. Later, discovering the generalisation of these to rings and other fields was all the more fascinating-not to mention the links to Linear Algebra. More recently, I have looked at more set-theoreticconstructions, in particular enjoying Zermelo-Fraenkel set theory, and the von Neumann hierarchy of sets. I took a fourth-year course in Axiomatic Set Theory to further endorse my interest in this area. My curiosity for mathematical discovery has nothing but intensified over my Masters degree, especially when it came to researching the separation axioms for a presentation during my second year. It stood out as my first taste into what researching for a dissertation would be like and taught me a lot about how to structure mathematical presentations. It is my drive towards self-conducted research and investigation that would form the basis for how I would undertake the research for thisproject, whilst further investigating Lie Theory and Representation Theory. I would love the opportunity to be able to further explore Lie Theory and Representation Theory, and to discover new and interesting results. I know that this project is the perfect vessel into which I can focus my curiosity for mathematical discovery and my passion for theoretical exploration. It will be another step towards my dream of giving back to the mathematical community by becoming a lecturer and researcher, and supporting the next generation of mathematicians. I have thoroughly enjoyed every stage of my mathematical development so far, and it excites me that working towards a PhD from the University of Birmingham would allow me to take this to a whole new level.