Inverse Problems in Geometry and Partial Differential Equations

点击翻译按钮获取中文摘要

DMS-0408419Title: Inverse problems in geometry and partial differential equationsPI: Peter A. Perry, University of KentuckyABSTRACTThis project involves inverse spectral and scattering theory in three areasof mathematical investigation: (1) the spectral theory of two-stepnilpotent groups and compact nilmanifolds, (2) the inverse resonance problem forexterior domains and scattering manifolds, and (3) inverse scattering forsingular potentials with applications to nonlinear dispersive equations.Two-step nilpotent lie groups play an important role in pure mathematics as models of sub-Riemannian geometry and as a rich source of examples ofmanifolds with identical Laplace spectra but distinct geometries; adetailed investigation of the trace formula for these manifolds will becarried out using analysis on nilpotent Lie groups. Resonances arediscrete scattering data for non-compact manifolds analogous to theeigenvalues of the Laplacian on a compact manifold; the inverse resonanceproblem will be investigated in the contexts of conformal andasymptotically flat geometries, and invariants such as the determinant willbe studied. Nonlinear dispersive equations with singular initial data maybe viewed, via the inverse scattering method, as linearized flows for thescattering data of a very singular potential. We hope to extend the inversescattering picture for the KdV and mKdV equations to such singular data andobtain greater insight into these dynamical systems--by constructing theflow on Hilbert spaces of initial data which are singular but very naturalfrom a dynamical point of view. Inverse spectral theory is the mathematical discipline thatunderlies important applications of mathematics to medical imaging,geophysical prospection, non-destructive testing, and many other areas.In these applications, properties of a physicalsystem (a human body, the earth, or an industrial material) are deduced from its response to externally imposed stimuli (electromagneticradiation, seismic waves, or ultrasound). The properties deduced may loosely be described as the "geometry" of the system and its response to external stimuli the "spectral data" (or "normalmodes"). A deep result of the study of completely integrable systems is that certain physical phenomena, such as the propagation of waves inshallow water, can be solved using an associated inverse spectral problem.Thus advances in inverse spectral theory lead to a better understanding ofhow such nonlinear waves propagate. The impact of this project will be twofold: first, it will elucidate, by studying carefully chosen geometric contexts, the relation between speectrum and geometry. Secondly, it will deepen our understanding of nonlinear dispersive waves by extending tools of inverse scattering theory to study nonlinear wave propagation with very singular waves.