My area of research is Arithmetic Algebraic Geometry, which is the common part of Number Theory, Algebra, and Geometry. I am very much interested in moduli spaces, group schemes, Lie algebras, formal group schemes, representation theory, cohomology theories, Galois theory, and the classification of projective, smooth, connected varieties.
My research is focused on:
- Shimura varieties of Hodge type (which are moduli spaces of polarized abelian varieties endowed with Hodge cycles),
- arithmetic properties of abelian schemes,
- classification of $p$-divisible groups,
- representations of Lie algebras and reductive group schemes,
- crystalline cohomology of large classes of polarized varieties,
- Galois representations associated to abelian varieties,
- arithmetic aspects over finite fields such as Waring problem for matrices and approaches to Jacobian Conjecture,
- arithmetics properties of special classes of rings such as Hermite rings, and
- arithmetic properties of affine algebraic geometry such as automorphisms of affine spaces and Jacobian Conjecture.
