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Finite Type Conditions in Harmonic Analysis

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Speaker: Andrew Comech, Duke University
Location: 693 Kerr
Start time: Mon, Feb 9 2004, 4:10PM

Methods of modern Harmonic Analysis are being constantly refined to adapt to the needs of the theory of Partial Differential Equations. The usual challenge is to relate the underlying geometry to the properties of solutions in the most accurate way. Two following situations are of a particular interest to us: (a) Restriction of a solution to the wave equation onto a hypersurface in space-time, in the case when this hypersurface is not strictly space-like (when it is time-like or even has characteristic points); (b) Blow-up of a solution to the wave equation at the caustic points. In each of these situations, certain standard assumptions break down, and one wants to know how badly this affects the result. More precisely, one wants to express the geometry of a problem in terms of certain finite type conditions, and then to relate these conditions to the loss of smoothness of solutions (compared to the smoothness in the "standard" situation). These questions are related to central research topics in modern Harmonic Analysis. We will discuss some recent results and open problems.