I am currently working under Prof. Terence Tao in the emerging field of arithmetic combinatorics, more traditionally known under the name of additive and combinatorial theory, with tools from analytic number theory, combinatorics, harmonic analysis and ergodic theory. This area is best described, as Gowers put it, as part of number theory where neither algebraic methods nor the Riemann zeta function and its generalizations play a central role. Most well-known results in this area include Szemerédi's theorem on the existence of arbitrarily long arithmetic progressions in dense subsets of the integers, and the Green-Tao theorem on the existence of arbitrarily long artihmetic progressions in the primes. For now, I am primarily working on questions in the function field model, since this is this is a nice object which shares a lot of similarities with the integers, and allows me to learn the many techniques of the field.

This list is still short, for now. I hope I will be able to make it longer.

• Sárközy's theorem in function fields, in preparation
• (with Craig Spencer) Difference sets and the irreducibles of Fq[t], preprint.
• Intersective polynomials and the primes, submitted.
• Green-Tao theorem in function fields, submitted.
• (with J. Cilleruelo) On a question of Sárközy on gaps of product sequences, to appear in Israel Journal of Mathematics

A couple of things I wrote in the past: 

• My undergraduate thesis on Preda Mihăilescu's proof of Catalan's conjecture, that I wrote under the supervision of Peter Stevenhagen at Universiteit Leiden, the Netherlands.
• My master's thesis (in French) on applications of pairings in Cryptography, that I wrote under the supervision of Jacques Stern and Phong Nguyen at the ENS Paris. 
• (This has nothing to do with math) My article on the Vietnamese language (in French), appeared in La jaune et la rouge, journal for Ecole Polytechnique alumni. Here is the English version, but don't count on the accuracy. I was tempted by languages and linguistics, but have never really thought about the idea of switching, since I am aware of the fact that, not getting one's hands dirty, one is usually attracted by what one is not actually doing :).

This list is even shorter, so please don't hesitate to help me expand it.

• Javier Cilleruelo, Universidad Autónoma de Madrid.
• Craig Spencer, Kansas State University.

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