Problems in Euclidean harmonic analysis related to the geometry of curves and surfaces

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The purpose of this project is to investigate the properties of a number of operators that play a central role in harmonic analysis. These are the restriction of the Fourier transform, averaging operators, the restricted X-ray transform and maximal operators. What unifies the study of the above is the presence, in each case, of some underlying curve or surface whose geometric properties (e.g. the curvature) determine the mapping properties of the corresponding operators. In many cases it is also natural to aim for estimates that are uniform over large classes of the underlying varieties. These uniform estimates are optimal in a certain sense. In order to achieve our goals, we will need to further our understanding of issues such as the decay properties of the Fourier transform in relation to Newton polyhedral, geometric inequalities involving an important quantity called the affine arc length and their interplay with certain combinatorial techniques.