Some of Mark's research interests.

Most of my current research centres on p-adic Hodge theory and its applications to the arithmetic of modular forms and modularity of Galois representations.

p-adic Hodge theory is a fairly young branch of number theory which seeks to study the p-adic analogue of the relationship between Betti cohomology and de Rham cohomology over the complex numbers. The p-adic analogue of Betti cohomology is p-adic etale cohomology, and it is related to other cohomology theories using the p-adic period rings introduced by J.M Fontaine. Since p-adic etale cohomology carries a Galois action, the theory contains a lot of rich arithmetic information.

p-adic Hodge theory has found many number theoretic applications. The most spectacular of these is perhaps Wiles' proof of the Taniyama-Shimura-Weil conjecture, and Mochizuki's work on Grothendieck's anabelian conjecture. It also plays an important role in Faltings' proof of the Mordell and Tate conjectures.

A recent exciting development in p-adic Hodge theory is Breuil's p-adic Langlands program, which relates (2-dimensional) p-adic Galois representations
and p-adic unitary representations of GL_2(Q_p). This nascent theory has already found applications in proving instances of modularity
of global Galois representations (the Fontaine-Mazur) conjecture.

p-adic modular forms

p-adic modular forms may be thought of as p-adic deformations or interpolations of classical modular forms. They may also be viewed of as global
avatars of the unitary representations of Breuil. One can often attach Galois representations (of the absolute Galois group of Q) to p-adic eigenforms.
Using p-adic Hodge theory one can show that the representations attached to finite slope, overconvergent eigenforms satisfy the Fontaine-Mazur conjecture.
Studying the p-adic periods of these representations yields a lot of information about the geometry of the space of such eigenforms (the "Eigencurve" of Coleman-Mazur).