Module MATH51760: Mathematical Sciences II

Department: Mathematical Sciences

MATH51760: Mathematical Sciences II

Prerequisites

• None

Corequisites

• Math Sciences I (MATH51860)

Excluded Combination of Modules

• None

Aims

• To acquire specialist knowledge in respect of several topics in Mathematical Sciences at MSc level, and to appreciate the highly rigorous and logical methods involved. In respect of the particular areas, to acquire ability in applying the theory and practice of this knowledge to standard and novel problems or in explaining fundamental aspects of the theory.

Content

• Students take unit A and, subject to the approval and with the advice of the course director, choose two units from B to J.
• (A) Further Topics: Two of the following topics will be offered: - A1) Global analysis and topology: Fixed point theorems and their applications, smooth manifolds, differential forms, de Rham cohomology; Morse theory; elements of singularity theory. - A2) p-adic theory: p-adic metrics, the field Qp of p-adic numbers as a completion of the field of rational numbers Q. Arithmetic in Qp. Hensel's Lemma. Quadratic forms with p-adic coefficients, the Hilbert symbol and equivalence of binary forms, Minkowski-Hasse theorem, comparison of non-Archimedean and Archimedean analysis, standard functions in the p-adic context. - A3) Lie theory: Lie groups and Lie Algebras; the correspondence between semisimple compact Lie groups and complex semisimple Lie algebras; the root space decomposition of semisimple complex Lie algebras; classification of complex simple Lie algebras and the corresponding compact simple Lie groups. - A4) Measure theory and dynamical applications: Abstract theory, (Sigma algebras, Borel Measures, measurable maps) Lebesgue measure. Birkhoff ergodic theorem, applications. Dynamical properties of maps. Examples and consequences. (e.g. Irrational rotations, power maps on the circle, the continued fraction map.)
• B) Algebraic Topology: Homotopy theory of cell complexes. Fundamental group. Covering spaces. Elements of homological algebra. Homology theory of topological spaces. Homotopy groups.
• C) Riemannian Geometry: The metric geometry of Riemannan manifolds. Geodesics. Various notions of curvature and the effect on the geometry of a Riemannian manifold. Second variation formula, global comparison theorems with applications.
• D) Topology: Topological Spaces and Continues Functions. Examples of topologies. Compactness and connectedness; Topological groups and group actions; orthogonal groups; the fundamental Group, calculation for circle, homotopy type, homotopy equivalence; generators and relations of groups; Tietze's theorem; van Kampen's theorem; compact surfaces; classical knots, basic knot invariants; Reading material on higher homotopy groups.
• E) Differential Geometry: Curves, surfaces in Rn; first fundamental form; mappings of surfaces; geometry of the Gauss map; intrinsic metric properties; Theorema Egregium; geodesics; minimal surfaces; Gauss Bonnet Theorem; reading material on the covariant derivative, the first and second variation formulae and the Bonnet-Myers theorem.
• F) Elliptic Functions: Elliptic integrals; general properties of elliptic functions. Weierstrass elliptic functions, and the expression of general elliptic functions in terms of the p-function and its derivatives. The addition theorem. Jacobi functions. Modular forms. Picard's theorem.
• G) Geometry: Isometries of the plane; affine transformations of the plane. Spherical geometry. Mobius transformations. Projective geometry; Desargues' and Pappus' Theorems and their duals; conics. Hyperbolic geometry; various model spaces, isometries, tessellations.
• H) Number Theory: Diophantine equations. Unique factorization and applications. Ideals; prime and maximal ideals. Principal ideal domains. Euclidean rings. Fields and field extensions. Algebraic integers. Quadratic fields and integers.
• I ) Partial Differential Equations: First order PDEs. Systems of first order PDEs, conservation laws. Hyperbolic systems; classification and reduction to standard forms. Parabolic equations; energy methods, maximum principles. Finite difference equations; stability and convergence for solutions. Iterative methods.
• J ) Approximation Theory and Ordinary Differential Equations: Numerical solutions of ODEs; Runge-Kutta methods, linear multiskip methods, predictor-corrector methods and error estimation, stability. Approximation theory; spline functions, minimax and near minimax polynomial approximations. Approximation by rational functions. Fast Fourier transforms.

Learning Outcomes

• Students will, in each of the units studied, have an understanding of the specialised and complex mathematical theory together with mastery of a coherent body of knowledge demonstrated through one or more topics from the following: Geometry, Topology, Elliptic Functions, Number Theory, Differential Equations.

• Students will develop highly specialised and advanced mathematical skills in the areas studied. They will be able to solve complex, novel and specialised problems, draw conclusions and deploy abstract reasoning and mathematical intuition, with minimal guidance.They will develop their mathematical self-sufficiency and be able to read and understand advanced mathematics independently.

• (1) Abstract reasoning, analytical thinking, problem solving, creativity and numeracy, written presentation of an argument.
• (2) The ability to learn actively and reflectively and to develop intuition, the ability to tackle material which is given both unfamiliar and complex.
• (3) Self-organisation, self-discipline and self-knowledge.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures and assigned reading indicate what is required to be learned and the application of the theory to examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Subject material assigned for private reading develops the ability to acquire knowledge and understanding without dependence on lectures.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and their lecturer on students' progress.
• The examinations assess the knowledge acquired and the ability to solve both standard and novel problems.
• The ability to solve these problems will show that the key skills have been developed. (Group (1) is tested directly in the problem solving and group (2) either directly or indirectly by the testing of the knowledge acquired. For group (3), a student who has acquired the knowledge and skills to succeed in this module will necessarily have had to develop the ability to organise and execute a programme of work and will discover aspects of and limits to his/her ability).

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

12 or more assignments (of problems) set and marked. (Other assignments set with solutions provided).

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University