Spectral problems in semi-classical analysis, wave and heat trace asymptotics and group actions on symplectic manifolds

半经典分析中的谱问题、波和热迹渐近以及辛流形上的群作用

• 批准号: 1005696
• 负责人: Victor Guillemin
• 金额: $ 32.9万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005696&HistoricalAwards=false
• 关键词: Spectral; problems; semi; classical; analysis

AbstractAward: DMS-1005696Principal Investigator: Victor GuilleminThe principal investigator proposes to study various inverse problems involving the semi-classical Schroedinger operator and, together with Alejandro Uribe and Zuoqin Wang, Zoll-type phenomena for these operators. He also proposes to study, together with Emily Dryden and Rosa Sena-Dias, an equivariant version of the Abreu conjecture: Does the spectrum of the Laplace operator on a toric orbifold determine the orbifold? In addition he proposes to investigate, together with David Jerison, Yves Colin de Verdiere, Steve Zelditch and Hamid Hezari, spectral properties of the wave trace for the Laplace operator on a convex domain in the plane in the vicinity of the perimeter, L. He also proposes to continue his current collaboration with Dan Burns on asymptotic properties of the spectral measures associated with Toeplitz operators on Kahler manifolds and to work with his students on a number of problems in equivariant symplectic geometry: generalizations of the Delzant theorem for Poisson manifolds and manifolds with degenerate symplectic forms and computations in K-cohomology for GKM manifolds. Finally, a long-standing project with Shlomo Sternberg is to write an on-line text book on functorial properties of symplectic manifolds with applications to semi-classical analysis.The overall theme of this proposal is "inverse problems" in spectral theory of which the most famous classical example is Mark Kac's "Can one hear the shape of a drum." What one "hears" in many of the problems above are discrete sets of spectral data: sophisticated versions of the vibrations of a drum, and what one wants to glean from this data are sophisticated versions of the shape of the drum itself. There has been a lot of progress on problems of this type over the last two decades, but there have been as well a number of cautionary counterexamples which show that further progress may require techniques that are still in their infancy. A large part of the focus of this proposal is developing such techniques.

AbstractAward：DMS-1005696首席研究员：维克托Guillemin首席研究员提出研究涉及半经典Schroedinger算子的各种逆问题，以及Alejandro Uribe和Zuoqin Wang，这些算子的Zoll型现象。他还建议研究，连同艾米丽德莱登和罗莎塞纳-迪亚斯，一个等变版本的阿布鲁猜想：频谱的拉普拉斯运营商的复曲面orbifold确定的orbifold？此外，他建议调查，连同大卫Jerison，伊夫科林德Verdiere，史蒂夫Zelditch和哈米德Zeldari，频谱特性的波迹的拉普拉斯经营者在一个凸域在平面上的周边附近，L。他还建议继续他目前的合作与丹伯恩斯的渐近性质的频谱措施与Toeplitz运营商的Kahler流形和工作与他的学生对一些问题的等变辛几何：推广的Delzant定理泊松流形和流形退化辛形式和计算在K-上同调的GKM流形。最后，一个长期的项目与什洛莫斯滕贝格是写一个在线教科书函性质辛流形的应用，以半经典分析。这一建议的总体主题是“逆问题”的谱理论，其中最著名的经典例子是马克卡茨的“可以听到的形状鼓。在上述许多问题中，人们“听到”的是离散的光谱数据集：鼓振动的复杂版本，人们想从这些数据中收集的是鼓本身形状的复杂版本。在过去的二十年里，这类问题已经取得了很大的进展，但也有一些值得警惕的反例表明，进一步的进展可能需要仍处于婴儿期的技术。这项建议的重点很大一部分是发展这种技术。