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


  • 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.


  • 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