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MATH4151: TOPICS IN ALGEBRA AND GEOMETRY IV

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Type	Open
Level	4
Credits	20
Availability	Available in 2023/24
Module Cap
Location	Durham
Department	Mathematical Sciences

Prerequisites

Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3 and including Complex Analysis II (MATH2011), Analysis in Many Variables II (MATH2031), Algebra II (MATH2581).

Corequisites

None.

Excluded Combinations of Modules

None.

Aims

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

Content

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.

Learning Outcomes

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 andadvanced mathematical skills in the following areas: Spatialawareness, abstract reasoning.

Key Skills:

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

Lectures demonstrate what is required to be learned and theapplication of the theory to practical examples.
Assignments for self-study develop problem-solving skills andenable students to test and develop their knowledge andunderstanding.
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 acquiredand the ability to solve complex and specialised problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total
Lectures	42	2 per week for 20 weeks and 2 in term 3.	1 Hour	42
Problems Classes	8	Four in each of terms 1 and 2	1 Hour	8
Preparation and Reading				150
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / Duration	Element Weighting	Resit Opportunity
three-hour examination		100

Formative Assessment

More information

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