Problems in geometric analysis

项目的焦点将是逆谱几何和分析尼尔流形上的黎曼几何和次黎曼几何。逆谱几何是研究黎曼流形的几何可以从谱数据中恢复的程度。戈登和韦伯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，并比较它们的局部和全局几何。他们和他们的合作者还将考虑环面上线丛上的薛定谔算符的谱和奥比诺德的谱数据。对于非紧致流形，相关的谱数据是散射共振和散射相位。戈登、韦伯和保罗将与彼得·佩里一起，研究使用相同的散射数据构建黎曼度量的可能性。在次黎曼几何领域，Pauls将继续致力于更好地理解卡诺群中的变分问题，重点是极小曲面的正则性，Heisenberg群最佳等周常数的计算，以前工作的扩展到更一般的卡诺群，以及与次拉普拉斯理论有关的问题(与Doyle，Gordon和Webb)。他还将继续与米克沃尔夫(莱斯大学)在调和映射理论中的两个基本问题上合作。研究人员将解决逆谱问题、逆散射问题和亚黎曼几何。逆光谱几何学植根于光谱学，即从发出的光或声的特征频率来理解系统的本质的问题。研究人员将考虑具有相同光谱的物体(平面域、球或球等黎曼流形)的各种结构，并将它们的几何形状进行比较，以确定未经光谱确定的特定几何性质。在势中粒子的量子力学描述中，人们区分了普通边界态和散射态，它们的波函数是不可归一化的，其能量可以假定为可能值的连续体。逆散射理论试图从粒子与势的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。研究人员将通过构造和研究具有相同散射共振的势来解决这个问题。亚黎曼测量学的研究是由大量的物理现象推动的，包括轮式机器人控制、卫星导航和稳定以及热力学方面的问题。在这个时间点上，人们对指导这些类型的控制现象的一般理论知之甚少。正因为如此，研究人员将重点放在更好地了解基本模型空间上费用最小化问题的解决方案上。具体地说，研究人员专注于在这些环境中构造面积最小化的曲面，以便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.