Language skills: French (written and spoken) at high school level.
At the end of this learning unit, the student is able to : | |
| 1 | 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”.
- The Riemann sphere: definition, compactness, automorphisms, behaviour of functions at infinity, meromorphic functions on the Riemann sphere.
- Construction of holomorphic and meromorphic functions: series of holomorphic and meromorphic functions, infinite products of holomorphic functions, examples.
- Elliptic functions and elliptic curves: complex tori, Abel's theorem, Weierstrass theory (Pfunction, zeta and sigma functions), Jacobi's theory (sn, cn and dn functions), associated elliptic curves and addition theorems, inversion problem of elliptic integrals.
- Applications: pendulum equation, Poncelet problem, Mordell-Weil theorem.
H. Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, Paris, 1961.
G.A. Jones, D. Singerman, Complex functions, an algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987.
