Module MATH51560: Geometry and Physics II

Department: Mathematical Sciences

MATH51560: Geometry and Physics II

Prerequisites

• None

Corequisites

• Geometry and Physics I (MATH51660)

Excluded Combination of Modules

• None

Aims

• To acquire specialist knowledge in respect of several topics in Applied Mathematics and Mathematical Physics at level 4 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 units below including A and/or B.
• A) Advanced Quantum Field Theory: 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.
• B) Solitons: 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.
• C) 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.
• D) Partial Differential Equations: First-order equations, conservation laws and Riemann invariants. Hyperbolic systems and discontinuous derivatives. Acceleration waves. Classification of general second order quasi-linear equation. Energy methods for parabolic equations. Well-posed problems. Maximum principles; finite difference solution to parabolic and elliptic equations, stability and convergence: Literative methods of solving Ax=b. Reading material set on one of the following topics: strict form of the maximum principles for parabolic equations; error estimates for numerical solutions of partial differential equations.

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: Classical and quantum Field Theory, ordinary and partial 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 practical 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 applications of logic and high level of rigour as well as feedback for the students and the lecturer on students progress.
• The examinations assess the knowledge acquired and the ability to solve 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 suceed 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