Local algebraic theory
Lubin and Oort conjectures, automorphism group of a field of formal series, Nottingham group, ramification, non abelian class field theory.

Global algebraic theory and K-theory
Galois groups, Galois module structure, K-theory of rings of integers, genus formula for wild kernels, K-theory and Iwasawa theory.

Archimedean analytic theory
Asymptotic study of arithmetic functions, distribution of divisors, friable integers.

Ultrametric analytic theory
Frobenius structure of p-adic differential equations, multiplicative semi-normes, continuous endomorphisms of Hadamard algebras.