Mathematics research: ergodic theory and dynamical systems, Diophantine approximation, and fractal geometry. 

Lately, Ramirez's research has focused in a branch of number theory called Diophantine approximation. Here, the basic goal is to find out the extent to which one can use rational numbers (numbers that can be expressed as fractions) to approximate irrational numbers (numbers that cannot be expressed as fractions). Diophantine approximation has fascinating connections to dynamical systems, where one studies the time-evolution of geometric objects under certain processes. Fractal geometry also plays an important role. For example, sets of real numbers with prescribed approximation properties tend to have fractal dimension. 

This interview for the Wesleyan Newsletter has a bit more.