I am currently a mathematics PhD student at the University of Queensland working under the supervision of Masoud Kamgarpour. My interests are primarily in representation theory, algebraic geometry and number theory.
A theorem of Hukuhara, Levelt, and Turrittin states that every formal differential operator has a Jordan decomposition. This theorem was generalised by Babbit and Varadarajan to the case of formal G-connections where G is a semisimple group. In this paper, we provide straightforward proofs of these facts, highlighting the analogy between the linear and differential settings.
The Birch and Swinnerton-Dyer conjecture provides a link between two, seemingly distinct objects associated with an elliptic curve. Since the original statement of the conjecture by Birch and Swinnerton-Dyer, there have been many refinements so that, in its present form, the conjecture is quite technical. The purpose of this thesis is to rigorously describe and define all the technical components of the conjecture.