I am interested in number theory and connections with logic.
Given an equation, one can ask what is the solution set of this
equation, say, over some "structure". Is it empty? How large can it
complicated can it be? I am interested in problems of this type when
the underlying structure has interesting arithmetic (whatever that
means) such as the
integers, the rational numbers or meromorphic functions. And
instead of solution sets of equations, I look more generally at
positive existentially definable sets over suitable languages.
Why positive existential formulas? because one can see them as an
abstraction of Diophantine problems. For instance, to say that "n is the
sum of two squares" one can say "there exist x and y such that
n= x2+y2 ".
Apparently unrelated to the previous topic, there is the problem of how rational points in varieties approximate targets inside the variety, a subject that people call Diophantine approximation. The abc conjecture can be seen a an important special case.
I often investigate connections between Diophantine approximation and logic aspects of number theory. For instance, I proved that the first order theory of meromorphic functions over a non-archimedean field of positive characteristic is undecidable, and I used the same argument to prove that the abc conjecture implies a solution to the Erdos-Ulam problem on rational distance sets.
In Chile I spent a lot of time working on Büchi's n-squares problem. In
Canada, I mostly worked on the abc conjecture. Both
problems remain open (as far as I know). Recently, I have been thinking about some new ways to attack the abc conjecture, in particular using Shimura curves.