Riemannian geometry (Nader Yeganefar)

Abstract

This course introduces the first concepts of Riemannian geometry, as metrics, connections and curvature. A special focus will be made on manifold of constant curvature.

Contents

• First definitions, examples.
• Connections and curvature
• Jacobi field, constant curvature spaces.
• Curvature -1 metrics on a topological surface.

References

• Gallot, Hulin, Lafontaine, 'Riemannian geometry', Springer-Verlag, Universitext, third edition 2004.
• do Carmo, 'Riemannian geometry', Birkhäuser

Lie theory (Frédéric Palesi, Jean Raimbault)

Abstract

The goal of this course is to give a thorough introduction to the basic notions and results on Lie groups and algebras. The course is roughly divided in two halves. The first describes the general setting and gives fundamental results, in particular the Lie algebra--Lie group correspondance. The second gives the basic notions of structure theory of semisimple complex Lie algebras, with some applications to semisimple Lie groups.

Contents

First part: Lie theory

• Abstract Lie groups, closed subgroups and homogeneous spaces
• Lie algebra and Lie correspondence
• Nilpotent and solvable Lie algebras and groups
• Semisimplicity and Cartan's criterion

• Classical groups and Lie algebras
• Cartan subalgebras of complex semisimple Lie algebras
• Representations of 𝔰𝔩₂
• Root decomposition and classification
• Real Lie algebras and groups

References

• Knapp, Lie groups, beyond an introduction, Birkhäuser.
• J. P. Serre, Complex semisimple Lie algebras, Springer.

Hyperbolic group theory (Thierry Coulbois)

Abstract

This lecture aims at introducing the geometric theory of groups with a special focus on Gromov hyperbolic spaces and groups. A special focus will be made on studying classical examples, with which we will try to illustrate the different combinatorial and geometric tools.

Contents

• Words and free groups
• Fundamental group and coverings
• Cayley Graphs and word distance
• Gromov hyperbolic spaces
• Quasi-isometry
• Hyperbolic groups

References

• Ghys, de la Harpe, 'Sur les Groupes Hyperboliques d’après Mikhael Gromov' Springer-Verlag, Progress in Mathematics, 1990
• Mark, Margalit, 'Office Hours with a Geometric Group Theorist', Princeton University Press, 2017

Elliptic curves and modular forms (Daniel Disegni, Joaquín Rodrigues Jacinto)

Abstract

The main goals of this cours are to provide an introduction to the arithmatic theory of elliptic curves and automorphic forms, in particular via the links between the two appearing when one views the modular surface as a moduli space for complex elliptic curves.

References

• Koblitz, Introduction to Elliptic curves and modular forms, Springer
• Serre, Cours d'arithmétique, PUF.
• Silverman, The Arithmetic of Elliptic Curves, Springer
• Milne, Elliptic curves
• Weil, Elliptic functions according to Eisenstein and Kronecker, Springer.
• Zagier, Elliptic modular forms, chapter in The 1-2-3 of Modular Forms, Springer.