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. We provide a new proof of this theorem by showing that every differential polynomial has a linear factorisation. The latter statement can be considered as a differential analogue of Puiseux's Theorem. Our approach makes clear the analogy between linear and differential operators thus making the proof more transparent.
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.