Inverse Problems in Geometry and Partial Differential Equations

这个项目涉及数学研究的三个领域的逆谱和散射理论：(1)两步幂零群和紧零流形的谱理论，(2)前域和散射流形的逆共振问题，以及(3)奇异势的逆散射应用于非线性色散方程。两步幂零李群在纯数学中起着重要的作用，作为次黎曼几何的模型和具有相同拉普拉斯谱但不同几何的流形的丰富的例子；利用对幂零李群的分析，对这些流形的迹公式进行了详细的研究。共振是非紧致流形的离散散射数据，类似于紧致流形上拉普拉斯的特征值；反共振问题将在共形和渐近平坦几何的背景下研究，不变量，如行列式将被研究。利用逆散射方法，将具有奇异初值的非线性色散方程看作是一个非常奇异势的散射数据的线性化流动。我们希望将KdV方程和mKdV方程的逆散射图像推广到这样的奇异数据，并通过在Hilbert空间上构造初始数据的流来更好地了解这些动力系统--从动力学的观点来看，这些初始数据是奇异的，但非常自然。逆谱理论是一门数学学科，它为医学成像、地球物理勘探、无损检测和许多其他领域的重要数学应用奠定了基础。在这些应用中，物理系统(人体、地球或工业材料)的性质是根据其对外界刺激(电磁辐射、地震波或超声波)的反应而推断出来的。所推导的性质可以粗略地描述为系统的“几何形状”及其对外部刺激的响应，即“光谱数据”(或“正常模式”)。完全可积系统研究的一个深刻结果是，某些物理现象，如浅水中的波的传播，可以用相关的逆谱问题来解决。因此，逆谱理论的进步导致了对这种非线性波如何传播的更好的理解。这个项目的影响将是双重的：首先，它将通过研究精心选择的几何背景来阐明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具有非常奇异的波的非线性波的传播，将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.