MATH41520: Topics in Algebra and Geometry

• Prior knowledge of Complex Analysis, Analysis in Many Variables and Algebra at undergraduate level.

• None

• None

• To introduce a contemporary topic in pure mathematics and to develop and apply it.

• One of the following topics:
• Elliptic functions and modular forms: to introduce the theory of multiply-periodic functions of one complex variable and the closely related theory of modular forms and to develop and apply it.
• Algebraic curves: to introduce the basic theory of plane curves, with a particular emphasis on elliptic curves and their arithmetic.
• Analytic number theory: to understand important results in analytic number theory related to the distribution of primes, in particular, the theory of the Riemann zeta function and Dirichlet series, gearing towards the proof of the prime number theorem. The course will demonstrate how to use tools from complex analysis to derive results about primes.
• Riemann surfaces: to introduce the theory of multi-valued complex functions and Riemann surfaces.

Subject-specific Knowledge:

• Ability to solve complex, unpredictable and specialised problems in pure mathematics.
• Understanding of a specialised and complex topic in theoretical mathematics.
• Mastery of a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: algebraic curves, elliptic functions and modular forms, analytic number theory, Riemann surfaces.

Subject-specific Skills:

• In addition students will have highly specialised and advanced mathematical skills in the following areas: Spatial awareness, abstract reasoning.

Key Skills:

• Lectures demonstrate 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.
• 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 end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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