Geometry

The mathematicians in this group (fr) work on the following topics:

Flat and negative curvature geometry

• Translation surfaces, Siegel-Veech constants.
• Discrete groups of isometries in negative curvature.
• Moduli spaces of Riemann surfaces.
• Representation varieties of surface groups.
• Rigidity of modular groups, Kleinian groups.

Topology in low dimensions

• Immersed surfaces in 3-manifolds.
• Classification of arithmetic link complements.
• Construction of hyperbolic manifolds.
• Modular groups, braid groups, Artin-Tits groups.

Foliations and differential equations

• Differential Galois Theory.
• Holomorphic foliations (local and global).
• O-minimal structures.
• Moduli space of connections.

Real and complex dynamics

• Groups of birational transformations of CP(2) and CP(3).
• Dynamics of laminations.
• Iterations of holomorphic maps.
• Group actions on the circle.
• Random dynamics on the real line.

The mathematicians in this group (fr) work on the following topics:

Non-Archimedean Geometry and Applications

Geometry of moduli problems

• Foundations of algebraic stacks, curves and their covers, torsors.
• Arithmetic applications.

Cohomological methods in arithmetic geometry

• Etale cohomology.
• p-adic cohomologies, especially crystalline and rigid cohomology
• Module theory over differential operator rings (D-modules) in positive characteristic

Arithmetic structures in p-adic Hodge theory

Representation theory of p-adic groups and the Langlands program

The mathematicians in this group (fr) work on the following topics:

Arithmetic and geometry

• Algorithms for p-adic numbers, in particular numerical stability issues.
• Effective aspects of arithmetic and geometry of abelian varieties, curves and their moduli spaces.
• Algorithmic aspects of real geometry.

Error Correcting Codes

• Use of the theory of twisted polynomials.
• Codes used in cryptography.

Cryptography

• Strengthening the security and effectiveness of existing protocols. This applies in particular to finite fields and elliptic curves (DLP problem, pairing, etc. ).
• Extension of the previous problems to higher genus.
• Exploring alternatives to based curve cryptography using codes.
• Study of random generators.

Differential Galois Theory

• Extension of the algorithms for finding Liouville solutions of linear differential equations to orders greater than 4.
• Algorithmic of operators and differential systems in characteristic p.

The mathematicans in this group (fr) work on the following topics

Motivic integration

• Geometry of arc spaces.
• Motivic Zeta functions, motivic nearby cycles and motivic monodromy conjecture
• Motivic Manin conjecture.

Motives

• Motivic cohomology.
• Motives and motivic stable homotopy theory.

Singularities of maps

• Stability.
• Stratifications.

Real algebraic geometry

• Semi-algebraic and o-minimal geometry.
• Real Milnor fibre.
• Positivity.
• Sums of squares
• Robotic.

Complex geometry

• Kähler geometry.
• Hodge theory.
• Ample and positive vector bundles.
• Kobayashi hyperbolicity.
• Geometry over complex functions fields.
• Birational geometry.
• Foliations.

History of mathematics