Here, you will find links to a selection of typed documents. The documents are organised by topic and are accompanied by a short description. Please notify me of any errors.
Algebraic Geometry
On June 2022, Dan Murfet and Ken Chan taught an eight week introductory course in algebraic geometry called MAG1 (Metauni Algebraic Geometry 1). Central to the course is the notion of a Gröbner basis. These are my typed notes regarding the course.
Category Theory
These are notes on the University of Oxford’s category theory course.
An introduction to protomodular categories
These notes provide a short introduction to protomodular categories. Protomodular categories possess an intrinsic notion of a normal subobject, which generalises the notion of a normal subgroup. I initially intended for these notes to contribute towards the broadening requirement for the DPhil degree. However, the requirement to submit a broadening project for each audited undergraduate course was removed after I created these notes.
C*-algebras
An introduction to operator theory
This document studies operators on a Hilbert space, beginning with bounded operators on a Hilbert space which is the prototypical example of a C*-algebra. In particular, the z-transform is introduced as a useful tool to study closed, densely defined operators on a Hilbert space.
This is a set of typed notes about the basic theory of C*-algebras. It serves as an introduction to the theory of C*-algebras, assuming that the reader has not encountered the notion of a C*-algebra before.
Examples of six-term exact sequences in K-theory
This document contains a few examples of six-term exact sequences associated to short exact sequences of C*-algebras.
Functional Analysis
Some notes on functional analysis
Constructed in 2019 with the initial intent of learning LaTeX, this document consists of material usually found in a first course on functional analysis. I plan on rewriting these notes so that the material is better communicated.
Integrable systems
Various aspects of the Ising model are discussed in this document.
Linear Algebra
Wedge product matrices are developed as a generalisation to the determinant of a square matrix with entries in a commutative ring. In tandem with a method Robert Steinberg used to prove a variant of the Bruhat decomposition, the two ideas were applied to various problems in linear algebra. Notably, they were used to generalise the eigenvector-eigenvalue identity and the notion of a quasideterminant in a non-commutative ring. They were also applied to the construction of representatives of two different matrix orbit spaces.