Approximation spaces of Boolean functions and their applications to complexity, pseudorandomness, learning, etc.
I am interested in continuing to develop my analytic framework based on approximation theory and other techniques. A number of apparently diverse complexity related questions -- on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean functions -- fit neatly into this framework. This isolates the analytic content of these problems from their combinatorial content and clarifies the close relationship between the analytic structure of questions.
This approach has been fruitful in providing new lower bounds, pseudorandom generators, learning algorithms, (reported in recent papers) and a new point of view for dealing with statistical tests; and I currently continue to pursue further challenges on all of these issues.
In fact, I would go further to say that the fundamental problems about the approximation properties of functions on the hypercube have to be answered before any significant progress can be made towards the P != NP problem. The nice thing is that these approximation problems are cleanly stated and have a basic appeal to mathematicians, especially approximation theorists or functional analysts.