Title: Differentiability of Fourier Restrictions

Abstract: We want to explore the connection between two well-known properties of the Fourier transform:

1) The Fourier transform of an integrable function may be continuous, but it typically is not differentiable at all.

2) If you look at the values of the Fourier transform along a surface, better things happen if the surface is curved instead of flat.

The end result is that we're able to extract some semblance of a derivative of the Fourier transform on curved surfaces, even when it's not differentiable in the classical sense.  In one special case we can show that the classical partial derivative actually exists at most points.

The talk will introduce some old and new results, and try to describe why they were all a bit surprising upon first discovery.

About the speaker: Michael Goldberg is the Head of the Department of Mathematical Sciences at the University of Cincinnati.  Prior to joining the faculty there, he held positions at the California Institute of Technology and Johns Hopkins University.  He received his Ph.D. in 2002 from UC Berkeley under the direction of F. Michael Christ. Professor Goldberg’s research interests are in Fourier analysis and its relationship to dispersive PDE, such as wave and Schrödinger equations.  His work has been supported by the Simons Foundation and the NSF.  Since 2011, he has also been a co-organizer of the Ohio River Analysis Meeting.