Module MATH51660: Geometry and Physics I

Department: Mathematical Sciences

MATH51660: Geometry and Physics I

Prerequisites

• None

Corequisites

• Geometry and Physics II (MATH51560)

Excluded Combination of Modules

• None

Aims

• To acquire specialist knowledge in respect of several topics in Pure Mathematics 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 (subject to the approval of the course director) unit A or unit B and choose two more units from units C, D, E, F and G (but F and G are excluded combinations).
• A) Riemannian Geometry: The metric geometry of Riemannian manifolds. Geodesics. Various notions of curvature and their effect on the geometry of a Riemannian manifold. Second variation formula, global comparison theorems with applications.
• B) Differential Geometry: Curves, Surfaces in Rn; first fundamental form; mappings of surfaces; Geometry of the Gauss map; Intrinsic metric properties; Theorema Egrgium; geodesics; minimal surfaces; Gauss Bonnet theorem. Reading material on the covariant derivative, the first and second variation formulae, the Bonnet-Myers theorem.
• C) 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.
• D) 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.
• E) 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.
• F) Algebraic Topology: Homotopy theory of cell complexes. Fundamental group. Covering spaces. Elements of homological algebra. Homology theory of topological spaces. Homotopy groups.
• G) 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.

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 and Number Theory.

• 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 study 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.
• 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