Dr. Erdogan will conduct basic research on harmonic analysis, geometric measure theory and partial differential equations, focusing on problems in harmonic analysis in Euclidean spaces centered around Lebesgue norm inequalities. One subject of ongoing research is the restriction estimates for the Fourier transform. Recently, the PI has obtained satisfactory results in some cases, which imply new partial results in the direction of Falconer's distance set conjecture -- a long standing open problem in geometric measure theory. These results also imply new Strichartz type estimates for the wave and Schrodinger equations. Another example of ongoing research is the mapping properties of Generalized Radon transforms (GRT) -- a huge class of averaging operators over lower dimensional sub-manifolds of Euclidean spaces. By applying the techniques of Bourgain, Wolff and others developed for Kakeya problems, he obtained (part of it jointly with Christ) interesting results in some cases. In recent years, the PI studied the dynamical properties of Schrodinger evolution (joint with Killip and Schlag). He will continue his investigations on various problems from mathematical physics, and the connections with harmonic analysis. Harmonic analysis has always found wide applications in natural sciences and engineering. It underlies a powerful and diverse array of tools currently widely used in applications, and offers the promise of further applications in the future. The proposed research deals with foundational issues, which may ultimately help to underpin such future applications. The study of the mapping properties of GRT has various applications in engineering. For example, the X-ray transform (which is a particular GRT) applied to the density function of a patients body is essentially the data obtained by magnetic resonance imaging. The study of GRT is also an important ingredient in the analysis of sumability of multi-dimensional Fourier series and integrals, of more general oscillatory integrals, and of a wide class of partial differential equations. The proposed research would make a contribution to the general understanding of these problems. The planned research is also related to certain discrete problems of interest in combinatorics and number theory, which in most cases remain wide open.
