*Note from June 29, 2020*

Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.

5 credits

30.0 h + 15.0 h

Q1

Teacher(s)

Haine Luc;

Language

French

Prerequisites

Complex analysis LMAT1222.

Language skills: French (written and spoken) at high school level.

Language skills: French (written and spoken) at high school level.

Main themes

Elliptic functions of Weierstrass and Jacobi, associated elliptic curves, Abel's theorem, addition theorem, selected applications in geometry, mechanics and number theory.

Aims

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Contribution of the course to learning outcomes in the Bachelor in Mathematics programme. By the end of this activity, students will have made progress in: - Recognise and undertsand a basic foundation of mathematics. In particular: -- Choose and use the basic tools of calculation to solve mathematical problems. -- Recognise the fundamental concepts of important current mathematical theories. -- Establish the main connections between these theories, analyse them and explain them through the use of examples. - Identify, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields. - Show evidence of abstract thinking and of critical spirit. In particular; -- Argue within the context of the axiomatic method. -- Recognise the key arguments and the structure of a proof. -- Construct and draw a proof independently. -- Evaluate the rigour of a mathematical or logical argument and identify any possible flaws in it. - Be clear, precise and rigorous in communicating. -- Write a mathematical text in French according to the conventions of the discipline. -- Structure an oral presentation in French, highlight key elements, identify techniques and concepts and adapt the presentation to the listeners' level of understanding. Learning outcomes specific to the course. By the end of this activity, students will be able to: - Construct holomorphic and meromorphic functions in terms of infinite series or products. - Apply Abel's theorem and the addition theorem of elliptic functions theory in various contexts. - Solve problems which use elliptic functions and elliptic curves. |

*The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.*

Content

In 2019-2020, we will study compact Riemann surfaces and their link with algebraic curves. Riemann was the first to understand that algebraic functions like "square root of z" are well defined and meromorphic on a branched covering of the complex plane with a point added at infinity. The concept was clarified only fifty years later by Hermann Weyl who introduced the notion of an abstract variety. We shall study the following topics:

1. Abstract Riemann surfaces, construction of the Riemann sphere.

2. Construction of the Riemann surface of an algebraic function, determination of its genus.

3. Elliptic curves (Riemann surfaces of genus 1), elliptic billiard problem.

Algebraic curve theory is one of the oldest part of algebraic and complex geometry and plays today an important role in several areas of mathematics and theoretical physics.

1. Abstract Riemann surfaces, construction of the Riemann sphere.

2. Construction of the Riemann surface of an algebraic function, determination of its genus.

3. Elliptic curves (Riemann surfaces of genus 1), elliptic billiard problem.

Algebraic curve theory is one of the oldest part of algebraic and complex geometry and plays today an important role in several areas of mathematics and theoretical physics.

Teaching methods

Learning activities consist of lectures which aim to introduce fundamental concepts, to explain them by showing examples and by determining their results, to show their reciprocal connections and their connections with other courses in the programme for the Bachelor in Mathematics.

Students are assigned problems at the begining of the semester, that they must solve and present during the problem sessions, according to a fixed schedule.

Students are assigned problems at the begining of the semester, that they must solve and present during the problem sessions, according to a fixed schedule.

Evaluation methods

Assessment is based on an oral examination relating to theory and on the individual assignment carried out during the term, in equal parts. The examination tests knowledge and understanding of fundamental concepts and results, ability to solve problems and ability to draft the solutions with rigour and clarity.

Online resources

The Moodle website provides a course outline, bibliographical references, as well as the problem sets to be done during the semester.

Teaching materials

- matériel sur moodle

Faculty or entity

**MATH**

#### Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme

Sigle

Credits

Prerequisites

Aims

Additionnal module in Mathematics