The course will start by introducing Riemann surfaces through an historical lens, with the study of conformal representation of embedded surfaces. The sequel will proceed with the formal definition and the study of several important classes of examples. The last part will be devoted to the proof of the existence of meromorphic functions on compact surfaces, and if time permit to its application to the algebraicity of Riemann surfaces.
This course is an introduction to the study of manifolds and complexes (CW and simplicial) from the viewpoint of algebraic topology. After recalling the necessary notions and result from general topology, the first part will introduce the fundamental group and its relation to coverings. The second part will introduce various homology and cohomology theories for manifolds and topological spaces, together with basics in homological algebra.
TBA
(Co)homology is a mathematical concept which produces simple invariants from algebraic or geometric structures of daunting complexity. These invariants encode essential information about these structures, which may be a topological space, a group or an algebra. All various (co)homology theories are built upon the common formalism of (co)chain complexes, the latter being built in appropriate fhasion from the particular object one is interested in.
The course will introduce the general theory and its applications :
Simplicial sets are combinatorial objects which which can be used to describe topological space up to "weak equivalence" by a result of Milnor. All notion from classical homotopy theory (for instance that of an homotopy between applications) can be given a pleasing definition in this combinatorial context.
Thus, modern homotopy theory tends to replace topological spaces with simplicial sets. This simplifie matters in various aspects: topologocal patholigies are no longer a concern, the category of simplicial sets is better behaved and new combinatorial tools can be applied. On the other hand, a significant loss from the topological setting occurs when one is faced with hard combinatorial problems.
Th goal of this course is to introduce simplicial sets and their homotopy theory. It will start with some categorical prerequisites on presheaves, in particular Yoneda's lemma. Then it will introduce simplicial sets and their skeleton decomposition. The realisation functor to topological spaces will next be presented, as well as its right adjoint and its application to singular homology. Milnor's theorem will be stated without proof. The last part of the course will be devoted to homotopy theory, in particular then notions of homotopy between simplicial morphisms, weak simplicial equivalence and Kan complexes, which lie at the basis of ∞-groupoid theory.