There are four research groups working in geometry: Analytic geometry, Arithmetic geometry, Computational geometry and algebra, Geometry and singularities.

## Analytic geometry

The mathematicians in this group (fr) [1] 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) [2] 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) [3] 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) [4] 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**