Topics in analysis related to resolution of singularities

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This project is at the interface of analysis and resolution of singularities, the latter area often being considered to be a part of algebraic geometry. Work will be done in several subjects, most of which have connections to various parts of mathematics. These subjects include restriction theorems for the Fourier transform and associated problems in partial differential equations, maximal averages over surfaces, and oscillatory integral operators with real phase. The investigator's earlier research has found applications in scalar oscillatory integrals with real phase and in stability problems for integrals, and another goal of this research is to provide further developments in these fields. The investigator is also interested in pursuing analogues of these oscillatory integral results to integrals over other fields such as the p-adics, which in turn are connected with various topics in number theory. Much of the research proposed is part of harmonic analysis (broadly construed), a field with substantial connections to applied fields such as wavelets, physics, and electrical engineering. In the long term, improved understanding of theoretical aspects of harmonic analysis has the potential to lead to applications in more applied subjects, which in turn have a broad impact on society as a whole. The investigator expects to have numerous interactions with students, helping in the development of the scientific human resources of this country. Furthermore, the investigator's university is an urban state university with one of the most diverse student populations in the country, having students of many ethnic and national origins. This occurs in both the undergraduate and graduate student body. Hence it is likely that working with students will help increase diversity in mathematics and related areas.