Mathematical Sciences: Metric Degeneration, Degenerating Elliptic and Parabolic Problems and Geometric Invarients of Elliptic Operators

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The principal investigator will study the geometric invariants of elliptic operators via the adiabatic limit method. In particular he will try to understand their behavior under a general process of surgery. One of the invariants to be studied is the analytic torsion. The eta invariant and determinant will also be considered. One motivation for the research is the recent work of Bismut and Lebeau on the arithmetic Riemann-Roch theorem. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.