Research in harmonic analysis and partial differential equations

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Erdogan will undertake mathematics research in the areas of harmonic analysis and partial differential equations (PDE). In harmonic analysis, Erdogan focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research are the restriction estimates for the Fourier transform. In recent years Erdogan and his Ph.D. student obtained satisfactory results in some cases. He proposes to continue his investigations on restriction estimates, and on their applications in PDE and geometric measure theory. In PDE, the focus is on the dynamical properties of dispersive PDE. One subject of on-going research is dispersive decay and smoothing estimates for the Schrodinger equation (SE). Another topic is the regularity of nonlinear dispersive PDE such as the Korteweg-de Vries (KdV) equation. Erdogan and his collaborators recently obtained smoothing bounds for KdV, and found applications to long-term behavior of the solutions. The techniques employed in this work are applicable to a wide range of PDE. Erdogan will continue his investigations in this direction. He will also work on mathematical problems on non-linear SE motivated by the numerical studies in fiber optic communication systems.Harmonic analysis has played major roles in pure and applied sciences since Fourier's seminal work on the theory of heat diffusion, continuing on with Schrodinger's equation in quantum mechanics. This mathematics research project is partly concerned with problems coming from real-world applications. For example, the X-ray transform, an integral operator, applied to the density function of a patient's body is essentially the data obtained by magnetic resonance imaging. The dispersion managed solitons are used commercially in transatlantic fiber optic internet cables. A better understanding of the underlying mathematical structure may be useful in improving these systems. The dispersive decay and smoothing estimates have a wide range of applications in nonlinear partial differential equations modeling diverse physical phenomena. In particular, the nonlinear Schrodinger equation models the transmission of data in fiber optic communication systems, and the Korteweg-de Vries equation models surface water waves as well as ion-acoustic waves in cold plasma.