Geometry

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

Some thematics studied by research groups in geometry
  1. Analytic geometry
  2. Arithmetic Geometry
  3. Computational geometry and algebra
  4. Geometry and Singularities

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