Tenth Prairie Analysis Seminar

If we have a real vector field in the plane, the integral curves foliate the plane. Similarly, if we have a complex vector field in C then the complex integral curves foliate space by Riemann surfaces. An example of a question one might ask, is whether these Riemann surfaces are dense in space. In the case of vector fields in the plane, this is impossible, but if we do it on a torus instead, it is easy. One method to do analysis on a Riemann surface is to use the current of integration. For a foliation one needs to average suitably over the individual Riemann surfaces to get good currents. We will explore these currents in these lectures. The work covered is mostly joint work with Sibony and Wold.