Analytic 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.
Arithmetic Geometry
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
Computational geometry and algebra
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.
Geometry and Singularities
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