This proposal deals with many of the central questions in Fourier analysis and how they manifest themselves on domains or more generally, Riemannian manifolds with boundary. In the context of the Fourier transform on Euclidean space or the flat torus, topics such as restriction theorems, Strichartz estimates (space-time integrability estimates for wave and Schroedinger equations), and local smoothing inequalities have been subjects of great interest for quite some time. However, many questions remain on how these theories should play out on a bounded or exterior domain. Here it may be appropriate to think of eigenfunctions of the Laplacian, and replace restriction theorems for the torus with integrability estimates on clusters of eigenfunctions. In this regard, the PI intends to explore estimates on the restriction of these clusters to curves in the domain. In the case of Strichartz estimates, the PI and his collaborators are using a parametrix-based approach to obtain results for general domains. Further improvement is expected in certain contexts by making use of a relatively new family of local smoothing estimates.
Fourier analysis continues to be a significant factor in the development of both mathematical and physical theories. In particular, it strengthens our understanding of the partial differential equations that arise in mathematical physics. The research pursued here expects to have several applications to the study of wave phenomena and the equations which model it. These investigations should yield insight on how the presence of a hard boundary surface influences the development of waves, a subject of great scientific interest.
The research pursued here provides the mathematical foundation for the study of light and sound waves and hence should be of broad interest in the study of wave propagation, both from mathematical and physical perspectives. Indeed, Fourier analysis continues to play a significant role in deepening our understanding of the equations which model wave behavior. In particular, these investigations yield further insight as to how the presence of a hard boundary surface influences the development of waves. For example, if one listens to the symphony in an auditorium, the sounds heard are affected by the manner in which the acoustic waves reflect off the walls. In this sense, it can be important to understand how the shape of the hall influences its acoustics. While this is of course a classical problem, there is more to be understood in terms of how these interactions influence dispersive properties. Moreover, this line of work is important in the analysis of closely related nonlinear equations arising from fiber optics and water waves, where there is much to be done in understanding and limiting the various types of instabilities which can occur. Other lines of research pursued here have close connections to discplines such as quantum mechanics and quantum chaos, where it is interesting to consider concentration phenomenon of vibrational modes. The funded research has also assisted the PI in mentoring graduate and undergraduate students. During the course of the project, the PI succesfully advised two female graduate students in their thesis work (one Ph.D. student and one Master's student) and one male undergraduate honors thesis. All three of these students worked on projects inspired by and closely related to the research performed by the PI. It is also signficant that the PI developed and taught a new graduate topics course during this period that filled in gaps in the graduate curriculum at the University of New Mexico. The PI has also been active in both organizing and participating in conferences and parallel sessions at regional meetings of the American Mathematical Society. Some of these have occurred at the University of New Mexico, which is important for exposing the students and faculty to current mathematical research and also brings added attention the research activity occuring at this institution. NSF funding has thus been pivotal in disseminating the PI's research and building relationships which have facilitated the success of these events.
