Inverse Problems in Geometry and Partial Differential Equations

几何反问题和偏微分方程

• 批准号: 0408419
• 负责人: Peter Perry
• 金额: --
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0408419&HistoricalAwards=false
• 关键词: Inverse; Problems; Geometry; Partial; Differential

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.

DMS-0408419TITLE：几何和部分微分方程的逆问题PI：彼得·A·佩里，肯塔基亚比亚大学大学项目涉及数学研究的三个领域的逆频谱和散射理论：（1）两步性的nilmanifolds和compact nilmanifolds的光谱，（1）歧管和（3）在非线性色散方程中应用的逆散射外向电势。两次尼尔氏谎言群在纯数学中作为亚里曼尼亚几何形状的模型以及具有相同的laplace laplace paspectra的模型，在纯数学中起着重要作用；对这些流形的痕量公式进行了研究，将使用对nilpotent Lie组的分析来解决。共振的非紧密歧管的弧形散射数据类似于拉普拉斯式的laplacian的散射数据。逆谐振问题将在共形和响应平坦的几何形状以及所研究的决定因素等不变性的背景下进行研究。具有奇异初始数据的非线性色散方程，可以通过反向散射方法查看，作为对非常奇异电位的散散数据的线性化流量。我们希望通过在初始数据的希尔伯特（Hilbert）空间上构造奇特的数据流，从而将KDV和MKDV方程的反截面图片扩展到此类动力学系统的奇异数据，并对这些动态系统进行更大的见解，这些数据是奇异数据的Hilbert Space，这些数据是奇异但非常自然的，从动力学的角度来看。 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).推导的属性可能会宽松地描述为系统的“几何”及其对外部刺激的响应“光谱数据”（或“正常码”）。对完全集成系统的研究的深刻结果是，可以使用相关的逆频谱问题来解决某些物理现象，例如波浪的传播。 该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环境，Speectrum和几何形状之间的关系来阐明。其次，它将通过扩展逆散射理论的工具来研究非线性波传播，以非常奇异的波浪来加深我们对非线性分散波的理解。