Complex geometry [ LMAT2260 ]

6.0 crédits ECTS 45.0 h > Schedule

Teacher(s)	Haine Luc ;
Language	French
Place of the course	Louvain-la-Neuve
Prérequis	Precursory courses : MAT 1222 Complex Analysis.
Main themes	- Abstract Riemann surfaces. - The Riemann surface of an algebraic function. - Sheaf cohomology and the Riemann-Roch theorem. - Baker-Akhiezer functions and equations of Korteweg-de Vries type.
Aims	The introduction by Riemann of a surface covering the complex plane on which an algebraic function w(z) solution of an irreducible polynomial equation P(z,w)=0 becomes single-valued and analytic, is at the basis of the modern concept of a variety. An abstract Riemann surface is defined as a complex connected one-dimensional variety. The aim of the course will be to establish that abstract compact Riemann surfaces are precisely the Riemann surfaces of algebraic functions. The proof will be done by using the tools of modern topology (covering spaces) and algebraic geometry (sheaf cohomology). We will also discuss the relation between compact Riemann surfaces and soliton theory.
Evaluation methods	Oral examination.
Teaching methods	Course : 3 h./week.
Bibliography	Support: - Otto Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer (1981) - Boris Dubrovin, Integrable Systems and Riemann Surfaces, Lecture Notes (preliminary version) (2009)
Other information
Cycle and year of study	> Première année de master [120] en sciences physiques, à finalité approfondie > Deuxième année de master [120] en sciences physiques, à finalité approfondie > Deuxième année de master [120] en sciences mathématiques, à finalité approfondie
Faculty or entity in charge	> MATH

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