November 21st and 22nd, the Department’s PDE Group held the second of three NSF supported graduate mini-schools in Partial Differential Equations for this academic year (information about the first mini-school is also available).
The main speaker was Chris Sogge of Johns Hopkins University, who spoke about pointwise bounds for eigenfunctions of the Laplacian on Riemannian manifolds. Fortunately, we had a great turn out for all the lectures as we drew a large audience of our own graduate students and undergraduates who have been working with Hans Christianson and Jason Metcalfe studying aspects of semi-classical analysis. A special thank you to Jason Metcalfe for arranging most of the details regarding the speaker and attendees, as well as to Elaine Bullock and Sunny Oakley from the math department staff for all the administrative support.
Chris’ first lecture on Thursday, November 21st outlined some of the aspects of spectral analysis and its connection in particular to the geodesic flow and the wave equation on a given manifold. He also outlined where some of the difficulties of extending the result to manifolds with boundary might arise. He focused on spectral cluster estimates in particular the first day. Then, on Friday, he was able to take the machinery he had introduced and walk the audience through the ideas of some very recent work he has done separately with Matt Blair of the University of New Mexico (who spent part of a sabbatical at UNC recently) and Steve Zelditch of Northwestern. The key arguments are to improve the pointwise bounds for spectral clusters if possible and to apply the Von Neumann ergodic theorem in order to relate the notion of properties of self focal points on the manifold to when one can improve a pointwise estimate on a spectral cluster.
In addition, there were two supporting lectures by Chris’ graduate students Hongtan Sun and Min Xue. Hongtan spoke Thursday about dispersive estimates for the wave equation on backgrounds with hyperbolic trapped orbits. Min spoke Friday about Strichartz estimates for Klein-Gordon equations in asymptotically Euclidean settings. In addition, we had graduate student Yuanlong Chen of University of Washington and postdoctoral fellow Peng Shao at the Institute for Advanced Study in attendance for the lectures.
All lecture notes and talk slides are available online at the mini-school web site. Congratulations to all involved for a second successful PDE mini-school. We look forward to the third.