 With Alex Wilkie he proved the decidability of real exponential fields (solving a problem of Alfred Tarski) modulo Schanuel's conjecture from transcendental number theory.With Lou van den Dries he initiated and studied the model theory of logarithmic-exponential series and Hardy fields.Together with David Marker and Lou van den Dries, he proved several results on the model theory of the real field equipped with restricted analytic functions, which has had many applications to exponentiation and O-minimality.