Inverse Problems in Geometry and Partial Differential Equations

DMS-0408419题目：几何与偏微分方程中的逆问题PI：Peter A.佩里，剑桥大学摘要该项目涉及三个数学研究领域的逆光谱和散射理论：（1）两步幂零群的谱理论和紧的幂零流形;（2）外部区域和散射流形的逆共振问题;（3）奇异势的逆散射及其在非线性色散方程中的应用。阶幂零李群作为次黎曼几何的模型在纯数学中起着重要的作用，并作为具有相同拉普拉斯谱但不同几何的流形的丰富实例来源;将使用幂零李群的分析来详细研究这些流形的迹公式。共振是离散散射数据的非紧流形类似的特征值的拉普拉斯算子在一个紧凑的流形，逆共振问题将在共形和渐近平坦几何的背景下进行研究，和不变量，如行列式将进行研究。具有奇异初值的非线性色散方程，通过逆散射方法，可以看作是一个非常奇异的势的散射数据的线性化流。我们希望将KdV和mKdV方程的逆散射图像扩展到这样的奇异数据，并对这些动力系统有更深入的了解--通过在Hilbert空间上构造初始数据的流，这些初始数据是奇异的，但从动力学的角度来看是非常自然的。 逆谱理论是一门数学学科，它是数学在医学成像、地球物理勘探、无损检测和许多其他领域的重要应用的基础。在这些应用中，物理系统（人体、地球或工业材料）的性质是从其对外界刺激（电磁辐射、地震波或超声波）的响应中推导出来的。推导出的性质可以松散地描述为系统的“几何形状”，以及它对外部刺激的“光谱数据”（或“正常模式”）的响应。完全可积系统研究的一个重要结果是，某些物理现象，如浅水中波的传播，可以用一个相关的逆谱问题来解决，因此，逆谱理论的发展使人们更好地理解这种非线性波是如何传播的。 这个项目的影响将是双重的：首先，它将阐明，通过研究精心选择的几何背景，频谱和几何之间的关系。其次，通过扩展逆散射理论的工具来研究非线性波的传播，从而加深我们对非线性色散波的理解。

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.