Mathematical Sciences: Degenerate Microlocal Methods in Geometric Analysis

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9626382 Mazzeo This proposal is in the general area of geometric and microlocal analysis. The investigator intends to continue the development of degenerate pseudodifferential techniques and to apply them to scattering theory and spectral analysis of elliptic partial differential equations. These techniques involve explicit geometric constructions of parametrics and generalized inverses for degenerate elliptic operators. Many problems in physics and geometry can be described by systems of partial differential equations. Elliptic partial differential equations - describing, for example, various heat flow phenomena - form a particularly well-developed branch within the theory of partial differential equations. The proposed research deals with the 'singular' aspect of elliptic differential equations which are theoretically more difficult than the smooth case.