Here, you will find links to selected documents regarding mathematics. 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
Category theory and internal structures
This document develops some of the basic notions of category theory with an emphasis on reflexive relations, equivalence relations and internal structures within finitely complete categories. In the final chapter, we briefly define unital and protomodular categories, which generalise particular observations about the categories of monoids, commutative monoids, abelian groups and groups.
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.
Matrices invariant under Lambda 2
This document highlights a small, but interesting question arising from the work in my MSc thesis.
A generalisation of the eigenvector-eigenvalue identity is developed and studied in this document. Again, this relies on material developed in my MSc thesis.
Miscellaneous
This document highlights a (hopefully) interesting method of computing a particular integral.