Mathematics 3000

Course outline:
The aim of this course is to introduce students to a selection of advanced topics in mathematics. MAM3000W is the full-year major course for the BSc degree. Credit for MAM3000W is obtained by selecting an approved combination of four modules from those listed below.
Such a selection must include at least one of the modules 3AL or 3MS. A student will not be given credit for MAM3000W without having completed the modules 2RA Real Analysis and 2IA Introductory Algebra. Students who did not take both these modules for MAM2000W will be allowed to take one of them as one of the modules for MAM3000W. Students who are given permission to do a second-year module as part of MAM3000W might be required to do additional reading and be examined on it. Written projects with oral presentations will be a component of this course.
While the modules listed below are included in this course, all the modules may not be offered in any one year. Each module consists of thirty lectures and twelve tutorials.

The syllabus covers the following topics:
3AL ALGEBRA An introductory course of modern abstract algebra involving the following concepts: algebraic operations; magmas and unitary magmas; semigroups; monoids; closure operators; equivalence relations; categories; isomorphism; initial and terminal objects; algebras, homomorphisms, isomorphisms; subalgebras; products; quotient algebras; canonical factorizations of homomorphisms; free algebras. Various classical-algebraic constructions for groups, rings, fields, and vector spaces, seen as examples of these concepts, will be described in tutorials.

3CA COMPLEX ANALYSIS An introduction to the theory of complex functions with applications.

3LC LOGIC AND COMPUTATION The propositional and predicate calculi: their syntax, semantics and metatheory. Resolution theorem proving

3MS METRIC SPACES An introduction to metric spaces and their topology, with applications.

3TA TOPICS IN ALGEBRA A selection from lattices and order, congruences, Boolean algebra, representation theory, naive set theory, universal algebra. (Please note that this module is not a prerequisite for entry to the Honours course in Algebra.)

3TN TOPICS IN ANALYSIS A selection from the implicit function theorem and inverse mapping theorem, Lebesgue integral, Fourier analysis, Hilbert spaces, Lebesgue and Sobolev spaces, Fractals and approximation theory. (Please note that this module is not a prerequisite for entry to the Honours course in Functional Analysis.)