Module MATH51860: Mathematical Sciences I

Department: Mathematical Sciences

MATH51860: Mathematical Sciences I

Prerequisites

• None

Corequisites

• Math Sciences II (MATH51760)

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 three of the following units subject to the approval and with the advice of the course director. A balanced selection of at least twelve of these units will be offered each year, and a list of these will appear in the MSc booklet.
• Independent Study: Students choose under the guidance of the course director a topic for independent study. For each topic reading material appropriate to independent study at level 4 is specified and this will define the material to be studied.
• Algebraic Topology IV: Homotopy theory of cell complexes. Fundamental group. Covering spaces. Elements of homological algebra. Homology theory of topological spaces. Homotopy groups.
• Riemannian Geometry IV: 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.
• Topology III: 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.
• Analysis IV: Metric spaces; differentiation on manifolds; integration on Euclidean n-space; differential forms; integration on manifolds.
• Differential Geometry III: 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.
• Galois Theory III: Rings and fields; field extensions; fundamental theorem of Galois theory. Galois extensions, general polynomial equations; finite fields; ruler and compass constructions.
• Elliptic Functions IV: 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.
• Algebraic Geometry IV: Plane curves; Bezout's theorem and applications; complex curves as real surfaces.
• Geometry IV: 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.
• Number Theory IV: 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.
• Representation Theory and Modules IV: Rings, ideals, modules; modules over principal ideal domains; similarity of marices; semi-simple algebras. Modules over a grup algebra; complex representations of groups.
• Partial Differential Equations IV: 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.
• Approximation Theory and Ordinary Differential Equations IV: 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.
• Advanced Quantum Theory IV: Action principles and classical theory, quantisation of free scaler fields; application to strings; Virasoro algebra; string contraints as generators of conformal transformations, representations, central charge; spectra: physical state condition, no-ghost theorem, critical dimension, open string spectrum; connection to gauge theory, non-abelian gauge symmetry and importance for the Standard Model. Closed string spectrum, connection to Gravity. Compactification, spinning string: gauge-fixed action, Ramond and Neveu-Schwarz boundary conditions, Super-Virasoro algebra, spectrum; Dirichlet branes.
• Solitons IV: Nonlinear wave equations. Progressive wave solutions. Backlund transformations for the Sine-Gordon and KdV equations. Conservation laws in integrable systems. Hirota's method. Nonlinear Shrodinger equation. The inverse scattering method. Toda equations.
• Continuum Mechanics IV: Fluid flows; Euler and Bernoulli equations; Navir-stokes equations. Compressible flows.
• General Relativity IV: Differences between special and general relativity. Differentiable manifolds, metric, covariant derivative, curvature. General relativity, black holes. Cosmology.
• Probability IV: Probability as a measure; probability under partial information; random variables; convergence. General theory; integration; limit results; choice of further topics.

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.

• 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 independent study in several units 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 the lecturer on students' progress.
• Summative examinations assess the knowledge acquired and the ability to solve both standard and novel problems.
• The ability to solve 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 may be 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