I work in model theory and some related areas. Most of my research, usually conducted in collaboration with others, falls under at least one of the following headings.
(1) Algebraic points on the graphs of transcendental functions (where, under appropriate assumptions, one proves that such points are, in some sense, scarce).
(2) Complex exponentiation (studying, from a model-theoretic point of view, the field of complex numbers equipped with the usual exponential function).
(3) NIP and Distality (certain model-theoretic tameness conditions).
(4) Generalised dense pairs of structures (structures which are in some way similar to the ordered field of real numbers equipped with a unary predicate for the subfield of algebraic real numbers).