Mathematical writing

MAS 165: Mathematics for physicists

Old web page for this course

Mathematics for computer science

Notes for computer science students, introducing modular arithmetic, linear algebra and some probability theory, with applications to RSA cryptography, computer graphics, Markov processes and the Google page rank algorithm.

View notes (PDF, 358KB)

Planar transformations

An addition to the linear algebra part of the above notes, on affine planar transformations.

View notes (PDF, 108KB)

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.

View introduction (PDF, 555KB)

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.

View notes (PDF, 559KB)

Real and complex analysis

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.

View notes (PDF, 466KB)

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.

View notes (PDF, 306KB)

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.

View notes (PDF, 339KB)

Measure theory and integration

These notes are a rapid account of measure theory and the Lebesgue integral, starting from first principles.

View notes (PDF, 229KB)

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.

View article (PDF, 72KB)

A brief review of the theory of symmetric spectra

This article is a very short summary of some aspects of the theory of symmetric spectra that I find useful in my work.

View article (DVI, 20KB)

Notes on partial differential equations

These are notes for a course I gave on partial differential equations in spring 2001 at Odense University. The file is in PostScript format.

View notes (PS, 482KB)


Email me at p.mitchener@sheffield.ac.uk if you have any questions.

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