Module Search

MATH4151: TOPICS IN ALGEBRA AND GEOMETRY IV

Please ensure you check the module availability box for each module outline, as not all modules will run in each academic year. Each module description relates to the year indicated in the module availability box, and this may change from year to year, due to, for example: changing staff expertise, disciplinary developments, the requirements of external bodies and partners, and student feedback. Current modules are subject to change in light of the ongoing disruption caused by Covid-19.

Type	Open
Level	4
Credits	20
Availability	Available in 2024/2025
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

Eight assignments to be submitted.

More information

If you have a question about Durham's modular degree programmes, please visit our FAQ webpages, Help page or our glossary of terms. If you have a question about modular programmes that is not covered by the FAQ, or a query about the on-line Undergraduate Module Handbook, please contact us.

Prospective Students: If you have a query about a specific module or degree programme, please Ask Us.

Current Students: Please contact your department.