Problems in geometric analysis

提案0306752 pi: Carolyn Gordon， Scott paul， David webb题目：几何分析中的问题项目的重点将是逆光谱几何和零流形上黎曼和亚黎曼几何的分析。逆光谱几何是研究黎曼流形的几何可以从光谱数据中恢复的程度。Gordon和Webb将考虑具有相同拉普拉斯特征值谱的紧致黎曼流形的构造，并比较它们的局部几何和全局几何。他们，连同他们的合作者，也将考虑薛定谔算子的频谱，对环面和轨道的光谱数据进行在线捆绑。对于非紧流形，相关的光谱数据是散射共振和散射相位。戈登、韦伯和保罗，以及彼得·佩里，将用相同的散射数据研究黎曼度量的可能构造。他们还将考虑薛定谔算子的等散射势和等散射障碍。在亚黎曼几何领域，paul将继续致力于更好地理解卡诺群中的变分问题，专注于最小曲面的正则性，海森堡群的最佳等周常数的计算，将以前的工作扩展到更一般的卡诺群，以及与亚拉普拉斯的谱理论相关的问题（与Doyle， Gordon和Webb联合）。他还将继续与MikeWolf （Rice University）合作，研究谐波映射理论中的两个基本问题。研究人员将解决反光谱问题，反散射问题，和亚黎曼几何。逆光谱几何植根于光谱学，从发射的光或声的特征频率理解系统性质的问题。研究人员将考虑具有相同光谱的物体（黎曼流形，如平面域、球或球体）的各种结构，并将比较它们的几何形状，以确定非光谱确定的特定几何性质。在量子力学对势中的粒子的描述中，人们区分了普通边界态和散射态，后者的波函数是非归一化的，其能量可以假定为可能值的连续体。反散射理论试图从与势相互作用的粒子所表现出的散射行为中尽可能多地了解势的性质。研究人员将通过构建和研究具有相同散射共振的电位来解决这个问题。亚黎曼几何的研究是由大量的物理现象驱动的，包括轮式机器人控制、卫星导航和稳定以及热力学问题。在这个时间点上，对指导这些类型的控制现象的一般理论所知相对较少。正因为如此，研究人员将专注于更好地理解基本模型空间上成本最小化问题的解决方案。具体来说，研究人员专注于在这些设置中构建面积最小化曲面，以揭示一些基本的几何原理，这些原理支配着这些类型问题的解决方案。

Proposal 0306752PIs: Carolyn Gordon, Scott Pauls, David WebbTitle: PROBLEMS IN GEOMETRIC ANALYISISThe focus of the project will be inverse spectral geometry andanalysis of Riemannian and sub-Riemannian geometries on nilmanifolds.Inverse spectral geometry is the study of the extent to which thegeometry of a Riemannian manifold can be recovered from spectral data.Gordon and Webb will consider constructions of compact Riemannianmanifolds with the same Laplace eigenvalue spectrum and compare theirlocal and global geometry. They, along with their collaborators, willalso consider the spectrum of Schroedinger operators on line bundlesover tori and spectral data for orbifolds. For noncompact manifolds,the relevant spectral data are the scattering resonances andscattering phase. Gordon, Webb, and Pauls, along with Peter Perry,will investigate possible constructions of Riemannian metrics with thesame scattering data. They will also consider isoscatteringpotentials for the Schroedinger operator and isoscattering obstacles.In the area of sub-Riemannian geometry, Pauls will continue workingtowards a better understanding of variational problems in Carnotgroups, focusing on the regularity of minimal surfaces, thecalculations of the best isoperimetric constant for the Heisenberggroup, extensions of previous work to more general Carnot groups andon problems related to the spectral theory of the subLaplacian (jointwith Doyle, Gordon and Webb). He will also continue working with MikeWolf (Rice University) on two fundamental problems in the theory ofharmonic maps.The investigators will address inverse spectral problems, inversescattering problems, and sub-Riemannian geometry. Inverse spectralgeometry is rooted in spectroscopy, the problem of understanding thenature of a system from the characteristic frequencies of light orsound emitted. The investigators will consider various constructionsof objects (Riemannian manifolds such as planar domains, balls, orspheres) which have the same spectra and will compare their geometryin order to identify specific geometric properties that are notspectrally determined. In the quantum mechanical description of aparticle in a potential, one distinguishes between ordinary boundstates and scattering states whose wave functions are nonnormalizableand whose energies can assume a continuum of possible values. Inversescattering theory seeks to understand as much as possible about thenature of a potential from the scattering behavior exhibited byparticles interacting with the potential. The investigators willaddress this problem by constructing and studying potentials with thesame scattering resonances. The investigations in sub-Riemanniangeometry are motivated by a wealth of physical phenomena includingproblems in wheeled robotic control, satellite navigation andstabilization and thermodynamics. At this point in time, relativelylittle is known about the general theory guiding these types ofcontrol phenomena. Because of this, the investigators will focus ongaining a better understanding of the solutions to cost minimizationproblems on basic model spaces. Specifically, the investigators arefocused on constructing area minimizing surfaces in these settings inorder to expose some of the fundamental geometric principles governingthe solutions to these types of problems.