Writing

On this page I put some mathematical notes not directly relevant to my research or current teaching, but that might be useful enough still to be worth making publicly available.
• MAS 165: Mathematics for Physicists
• The old web page for this course.
• The old web page for this course.
• Mathematics for Computer Science

• Notes from a gave to computer science students, introducing modular arithmatic, linear algebra and some probability theory, with applications to RSA cryptography, computer graphics, Markov processes and the Google Page rank algorithm.
• Planar Transformations

• An addition to the linear algebra part of the above notes, on affine planar transformations.
• A Rapid Introduction to Differential Geometry

• The definition of smooth manifolds, vector bundles and construction of the tangent bundle, critical values, vector and tensor fields (including Sard's theorem and the proof of the Brouwer fixed point theorem), some Riemannian geometry and geodesics, and finally differential forms and de Rham cohomology, ending with a proof of Poincare duality.
• Notes on K-theory

• In these notes, we give an account of the construction of the K-theory of graded C*-algebras, and its fundamental properties.
• Real and Complex Analysis

• These are notes on a course I gave for three years at the University of Sheffield. The course covers the basics of normed spaces, differentiation in higher dimensional spaces, the relation between real and complex differentiability, and exactness of differential 1-forms. The fundamental group and covers of spaces are also introduced, and the ideas are applied to complex analysis.
• Notes on Galois Theory

• A fairly rapid account of Galois theory, complete with exercises, starting from elementary principles. Not much more than the basics of group theory and linear algebra, and the definition of rings and fields are needed.
• Non-Commutative Probability Theory

• Mostly self-contained notes on free probability theory, starting from first principles. Topics covered include the free central limit theorem, random matrices, and the R- and S-transformations.
• Measure Theory and Integration

• These notes are a rapid account of measure theory and the Lebesgue integral, starting from first principles.
• A Primer on some Methods in Homotopy Theory

• This article is an exposition of the theory of simplicial sets and spaces, classifying spaces, homotopy colimits, the plus construction and the group completion theorem. The techniques are applied at the end of the article to compare constructions in algebraic K-theory.
• A brief review of the theory of Symmetric Spectra