Notes from courses I took as an undergraduate student at The University of Birmingham between 1997 and 2001. They contain errors so be careful!
Covers variation of parameters, reduction of order, and Frobenius' methods for ordinary differential equations. Introduces Legndre polynomials and Bessed functions, and integral transforms.
Download MSM2G2 Advanced Calculus (PDF, 255k).
Sigma algebras and probability spaces, Ito integration, stock pricing, Black-Scholes economic model, option pricing.
Download MSMYGF Mathematics Of Finance (PDF, 246k).
Rings and integral domains, ideals and quotients, polynomial rings, irreducibles, Euclidean domains.
Download MSMXP5 Polynomials And Rings (PDF, 190k).
Groups of matrices, groups of permutations, disjoint cycle notation, rotations and reflections. Cyclic groups, the symmetric groups, the dihedral groups, the general linear groups. Group axioms, subgroups, homomorphisms and isomorphisms, Cayley's theorem. Odd and even permutations. Group actions, the orbit-stabiliser theorem, cosets and Lagrange's theorem. Normal subgroups, the homomorphism theorem.
Download MSM2P2 Symmetry And Groups (PDF, 258k).
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Download MSMYP5 Group Theory (PDF, 210k).
Error-correcting coding.
Download MSMYP7 Coding Theory (PDF, 223k).
Mostly related to the 4-colour theorem.
Download MSMYPC Combinatorics (PDF, 4096k).
Complex representations, and in particular complex characters.
Download MSM4P4 Representation Theory (PDF, 271k).
Download MSMYP2 General Topology (PDF, 218k).
Download MSMYP3 Functions Of Several Real Variables (PDF, 219k).
Introduction to complex variable theory defines derivatives and integrals for functions in the complex plane and proves the ML Result and Cauchy-Goursat amongst other theorems. Examining series expansions leads to results on singularities and residues and their applications to evaluating real integrals.
Download MSMXG4 Complex Variable Theory (PDF, 251k).