Objectif
The main purpose of this research project is to use a multidisciplinary approach to attack some important open problems in Algebraic Geometry, Arithmetic Geometry , Quadratic Forms and Algebraic Groups
Algebraic geometry and the Theory of Motives
- generalization of the Suslin-Gabber rigidity theorems.: to give a purely field theoretic description of field extensions satisfying rigidity and to obtain results for henselian valuation rings
- proving results on the Nilpotency Conjecture in Voevodsky's triangulated category of motives and on finite dimensionality of motives ,These conjectures are related with the Conjectures of Beilinson Bloch and Murre on the existence of a suitable filtrations on Chow groups. For a complex surface of general type with p_g =0 , Bloch's Conjecture is equivalent to the finite dimensionality of the motive
- cohomological operations for algebraic cobordisms of smooth projective varieties.
- study of the category of motives modulo numerical equivalence by means of the realization of the latter as a full subcategory in the category of semi-simple admissible representations of the automorphism group of certain algebraically closed extension of the base-field.
- Study of the discrete invariants of a quadric by means of the Chow groups and the algebraic cobordisms of the respective quadratic grassmannians
Arithmetic geometry
- Ramification Theory for arithmetic surfaces. Number of points over a finite field
- Applications of the classification of formal groups and finite group schemes to good and semistable reduction of Abelian varieties in order to find explicit formulas for generalized Hilbert symbols over formal group modules over ramified multi-dimensional local field
- Construction of the affine Grassmanian for two-dimensional local field in order to find a connection between the Krichever correspondence for algebraic surfaces and the higher KP-systems
Quadratic forms
The research will develop along the lines of the results recently obtained by Vishik, Voevodsky, Karpenko and Rost on the classification of anisotropic quadratic forms .The structire of the Chow motive of a quadric will be further studied and the Vishik character. Bott periodicity in hermitian K-theory for any ring and its relation with the classification of quadratic forms).
Linear algebraic groups
The aim is to prove Rosenberger's Conjecture on the Tits alternative for generalized triangle groups; Also the theory of sums of orbits of algebraic groups will be developed in order to prove an analogue of Deligne-Simpson problem.
Thème(s)
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PARIS
France