Problems in geometric analysis

提案0306752 PI：Carolyn Gordon、Scott Pauls、大卫韦伯职务：几何分析中的问题该项目的重点将是反谱几何和黎曼和亚黎曼的分析。黎曼几何在nilmanifold。逆谱几何是研究黎曼流形的几何可以从谱数据恢复的程度。Gordon和Webb将考虑具有相同性质的紧致黎曼流形的构造。拉普拉斯本征值谱，并比较了它们的局部几何和整体几何。 他们，沿着与他们的合作者，也将考虑谱的Schroedinger算子的线的spectrum的tori和谱的orbifolds数据。 对于非紧流形，相关的谱数据是散射共振和散射相位。 Gordon、Webb、Pauls和Peter佩里将沿着研究用相同的散射数据构造黎曼度量的可能性。 他们还将考虑isoscatteringpotential的薛定谔算子和isoscattering的障碍。在该地区的次黎曼几何，保罗将继续致力于更好地了解变分问题的卡诺集团，专注于最小曲面的规律性，thecalculations的最佳等周常数的海森堡集团，扩展以前的工作，更一般的卡诺集团和问题有关的频谱理论的subLaplacian（jointwith道尔，戈登和韦伯）。 他还将继续与MikeWolf（莱斯大学）合作研究调和映射理论中的两个基本问题：逆谱问题、逆散射问题和次黎曼几何。 逆谱几何学起源于光谱学，即从发出的光或声的特征频率来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的各种构造（黎曼流形，如平面域、球或球体），并将比较它们的几何形状，以确定光谱不确定的特定几何性质。 在量子力学对势中物体的描述中，人们区分了普通的束缚态和散射态，散射态的波函数是不可归一化的，其能量可以假设为一个连续的可能值。 反散射理论试图从粒子与势相互作用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势来解决这个问题。 亚黎曼几何的研究是由大量的物理现象引起的，包括轮式机器人控制、卫星导航和稳定以及热力学等问题。 在这一点上，相对较少的是知道的一般理论指导这些类型的控制现象。 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的成本最小化问题的解决方案。 具体来说，调查人员专注于在这些设置中构建面积最小化表面，以揭示一些基本的几何原理，这些问题的解决方案。

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.