Kakeya Maximal Operators and Oscillatory Integrals

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ABSTRACT Tao will study geometrical analysis conjectures and results, and relate them to oscillatory integral statements such as the restriction conjecture. A typical example of the former is the Kakeya conjecture, that a set which contains a line segment in every direction in R^n must have dimension n. A typical example of the latter is the generalized Strichartz estimate for various PDE (the wave equation, the Schrodinger equation, etc.). It has been known since the 1970s that there is an intimate relationship between the two types of statements, but no systematic approach exists, and the few concrete connections that are known are not very satisfactory. We hope to collect, simplify, unify, and extend previous results in this direction. Tao will also attack some of these conjectures (notably the Kakeya conjecture) directly, using some new and promising techniques; for example, Tao will exploit the affine invariance of the Kakeya conjecture. One of the aims of this work is to deepen our understanding of oscillatory integrals, which are a type of mathematical expression which occur in many places in physics (optics, quantum mechanics, acoustics, and any other field of physics dealing with waves), as well as having theoretical importance in other fields of mathematics. Understanding these integrals, and in particular knowing how large they can get, may ultimately lead to new designs for physical applications (e.g. tennis rackets that maximize the area of the "sweet spot", or curved reflectors that have a large number of focus points for a wide range of frequencies), or at least place theoretical limits on such designs. There are also numerical applications when modeling certain physical systems (e.g. the seismic behavior of the Earth); if one knows that a certain oscillatory integral will never become very "large" in a certain technical sense, then this will provide a theoretical guarantee to the accuracy of the computer sim ulation of the physical system. Tao will study these oscillatory integrals with the aid of geometry; a connection between these two fields of mathematics is known (being somewhat similar to the relationship between geometrical optics and the wave theory of light), but is not understood completely. If this connection is developed thoroughly enough, we may be able to reduce difficult questions in oscillatory integrals to simpler problems in geometry, or at least use geometrical techniques to obtain partial progress on the oscillatory integral problems.