• Semialgebraic and subanaytic geometry
• Real singularities
• Algorithms in real algebraic geometry
• Positivity, sums of squares
• Differential Galois theory. 
• Hamiltonian mechanics
• Cryptography
• Error-correcting codes
• Robotics 
• Epistemology and history of mathematics

In France

• ANR SIngularités REelles (Real singularities)
• ANR SIngularités des RObots PArallèles (Singularities of parrallel robots)
• ANR Courbes Hyperelliptiques Isogénies et Comptage (Hyperelliptic curves, isogeny and point counting) 
• GDR Singularités

International

• Réseau de Géométrie et Algèbre Appliquées au Développement, Soutien aux Activités de Recherche Informatique et Mathématique en Afrique

Presentation of the group:

The activity of the research group has been for many years focussed on real algebraic geometry and Rennes is a recognized centre in this theme. Four young associate professors or researcher got professor positions in other universities in the recent years. The study of algorithms in real algebraic geometry initiated the development and the diversification of the activities in symbolic computation. This led to scientific contacts with researchers in cryptography at CELAR, now associate researchers in IRMAR; the development of this research theme and the hiring of a professor working in this domain allowed to launch a master curriculum in mathematics of information and cryptography. Our group also hosts activities in history of mathematics.

The resarch activities are intere related with the ones of the other groups in geometry. The strong points of our group are essentially :

1. Real geometry, and specifically real singularities

• The study of real singularities offers many interesting and difficult questions and is very active now ; a central theme is the study of the real Milnor fibers, smooth pertubations of the singular object carrying carrying relevant information on it; this is the subject of the ANR contract SIRE (Real singularities) headed by Rennes.

• Real singularities are also important for applications, such as robotics. Rennes is part of an ANR contract SIROPA (Singularities of parallel robots) headed by the Ecole Centrale in Nantes.
  
  

2. Cryptography and error-correcting codes
   
• Public key cryptography is widely used in everyday's life. and the systems based on elliptic curves are now well-known. Hyperelliptic curves offer interesting perspectives. The study of their possible use for cryptography is the subject of the ANR contract CHIC (Hyperelliptic curves, isogeny and point counting) headed by Rennes.
  
• Non commutative algebra offers new ways to construct interesting error correcting codes. This is developed in the project « skewcode » .