PMA Level 2 Modules
Pre-requisite: MTH1001 and MTH1002
Lecturer: Dr B McMaster
This module extends and develops five of the key ideas from MTH1001, namely convergence of sequences and of series, and continuity, differentiation and integration of real functions.
Cauchy sequences, especially their characterisation of convergence. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted). Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Mean value theorems including that of Cauchy, proof of L'Hôpital's rule, Taylor's theorem with remainder. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.
Exam 60% Class Test 15% Projects 15% Continuous Assessment 10%
L. Alcock, How to Think About Analysis (Oxford University Press)
R. Haggarty, Fundamentals of Mathematical Analysis (Addison-Wesley)
A. McCluskey and T.B.M. McMaster, Undergraduate Analysis - a Working Textbook (Oxford University Press)
M. Spivak, Calculus (Cambridge University Press)
Lecturer: Prof I Todorov
Consider the symmetries of a square. It has four rotations: by π/2, πi, 3πi/2 and 2πi, which is the identity transformation! It has four reflections: horizontal, vertical and two diagonals. Any combination of these symmetries is another such symmetry and each symmetry has an inverse that undoes the transformation. This is an example of a group, a mathematical abstraction of a collection of symmetries of an object.
Another example of a group is the set of bijections of the numbers 1 to n. This group is called the symmetric group on n letters, it is fundamental to the study of groups and was one of the earliest studied examples. Permutation groups are used throughout mathematics and were used in the search for a general formula for the roots of a polynomial. They can also be used to describe card shuffles.Overall, group theory (along with linear algebra) is the basis of abstract algebra, a central theme of both pure and applied mathematics.
Equivalence relations, binary operations, groups, examples and properties, groups of maps, countability, homomorphisms, subgroups, equivalence relations, permutation groups, normal subgroups, quotient groups, structure of finite abelian groups, composition series and solvable groups.
Exam 80% Team Project 20%
MTH2001 Linear Algebra and Complex Variables (1st and 2nd semester)
Pre-Requisites: MTH1001 and MTH1002
Lecturers: Dr A Blanco, Prof J Kohanoff
Basic theory: matrices and Gauss elimination, LU decomposition; abstract vector spaces and subspaces, linear independence, basis, dimension; vector spaces associated to a matrix; general solution to inhomogeneous systems of linear equations; determinants; matrix inversion and computation of determinants; direct sums and direct products; inner products, orthogonality, Cauchy-Schwarz inequality; matrix representation of linear maps, eigenvalues and eigenvectors of matrices and endomorphisms; spectrum, diagonalization, similarity transformations; special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties
Additional topics and applications, such as: least-squares approximation, Schur decomposition, orthogonal direct sums, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form
Basic theory of complex variables: algebra of complex numbers, de Moivre's theorem, integer powers and roots; sequences of complex numbers, convergence, limits; series of complex numbers, convergence and absolute convergence, convergence tests; limits of functions, continuity; complex differentiability, analytic functions; power series; elementary functions: exp, sin, cos, sinh, cosh, log, complex powers; complex integration: paths, length of path, path integrals, fundamental theorem of calculus, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem, Taylor's theorem; Laurent series, Laurent's theorem, classification of isolated singularities, residues, Cauchy's residue theorem; computation of real integrals with complex methods
Additional topics and applications, such as: analytic functions as conformal maps, solutions to Laplace's equation, Moebius transformations, Cauchy-Riemann equations, harmonic functions, maximum modulus principle, Rouche's theorem, isolated zeroes theorem, fundamental theorem of algebra.
One 3-hour written examination 80%, 2 x homework assignments 10% .