Algebra

NUMBER THEORY AND ALGEBRAIC GEOMETRY

1) Study of the number of solutions of polynomial equations in several variables over finite fields and residue class rings, with emphasis on the relation to geometric and topological invariants of the equations, Igusa's local zeta function, p-adic exponential sums.

2) Connections between (1) and the theory of singularities, resolution of singularities, Log Minimal Model Program, topological and motivic zeta functions, monodromy,  Hodge spectrum, germs of arcs on an algebraic variety, asymptotics of oscillating integrals, principal value integrals of Langlands.

3) Monodromy conjecture via non-archimedean geometry, degenerations of algebraic varieties, connections with Mirror Symmetry via tropical geometry and logarithmic geometry.

4) Topology of algebraic varieties, local systems, perverse sheaves, mixed Hodge modules, regular and irregular D-modules, Bernstein-Sato ideals, resonance varieties.

5) Deformation theory, differential graded Lie algebras and modules, L-infinity algebra and modules, with applications to topology of algebraic varieties and to Brill-Noether theory of sheaves on algebraic varieties.

6) Connections with mathematical logic and model theory.

GROUP THEORY AND TOPOLOGY

1) Geometric group theory, almost-crystallographic groups, automorphisms of virtually nilpotent groups and more general groups, nilpotent and solvable Lie groups and Lie algebras, left symmetric algebras.

2) Nilmanifolds and infra-nilmanifolds, fixed point theory, Anosov diffeomorphisms.