Spectral analysis of geometric shapes

几何形状的光谱分析

• 批准号: 1001071
• 负责人: Mihai Putinar
• 金额: $ 19.79万
• 依托单位: University of California-Santa Barbara
• 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 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001071&HistoricalAwards=false
• 关键词: Spectral; analysis; geometric; shapes

The project is aimed at unifying three distinct matrix models (one quantum, the other random/thermodynamic and the third purely algebraic) known today in connection with planar elliptic growth, an interface dynamics covering a large variety of natural and theoretical phenomena. The existence of exact solutions to the equations governing such growth dynamics, their complete integrability, at least in the case of a flat metric, and the impressive amount of converging recent research on these topics make the project timely and unavoidable. As mathematical tools, the proposal will rely on and develop new facets of complex orthogonal polynomials, moment matrices and potential theoretic operators arising in the study of planar or spatial shapes. Particular emphasis will be put on the maximum entropy method for recovery of shade functions from indirect measurements, such as partial geometric tomographic data and the asymptotics of the spectra of random non-hermitian matrices.Following a decade of intense and vibrant discoveries in fluid mechanics and quantum physics, the present proposal addresses a series of mathematical questions of high interest for these specific areas of modern science. The tradition of encoding planar shapes and the geometry of volumes into numbers or algebraic symbols goes back to the XVII-century landmark contributions of Rene Descartes and his followers in what is today called ?analytic geometry?. Much later development and applications of electricity, magnetism and atomic science we all benefit today was possible by a second major step into the same direction, that is the representation of complex physical entities as large arrays of numbers or symbols called matrices. The proposed project focuses on the study of matrix models arising in the current research of moving interfaces of fluids or more sophisticated but similar media. Such moving boundaries phenomena are illustrated by cancer growth, crystal formation, ice melting, oil reserves, carbon monoxide sequestration and plasma dynamics. The PI is assisted by a group of enthusiastic doctoral students and he is connected, via several collaborative works, with experts in other fields of mathematics, physics or engineering.

该项目的目的是统一三个不同的矩阵模型（一个量子，其他随机/热力学和第三个纯代数）与平面椭圆生长，一个界面动态覆盖了各种各样的自然和理论现象。控制这种增长动力学的方程的精确解的存在，它们的完全可积性，至少在平坦度量的情况下，以及最近对这些主题的研究的令人印象深刻的收敛量，使得该项目及时和不可避免。作为数学工具，该建议将依赖于和开发新的方面复杂的正交多项式，矩矩阵和潜在的理论运营商在研究平面或空间形状。特别强调的是，将放在最大熵方法恢复的阴影功能，从间接测量，如部分几何断层数据和随机非厄米矩阵的谱的渐近性。继十年的激烈和充满活力的发现，在流体力学和量子物理学，本提案解决了一系列的数学问题的高度感兴趣的这些特定领域的现代科学。将平面形状和体积几何编码成数字或代数符号的传统可以追溯到十七世纪勒内·笛卡尔及其追随者的里程碑式贡献，即今天所谓的“几何学”。解析几何？我们今天都受益的电、磁和原子科学的发展和应用，可能是朝着同一方向迈出的第二个重大步骤，即把复杂的物理实体表示为称为矩阵的大量数字或符号。拟议的项目侧重于研究目前对流体或更复杂但类似的介质的移动界面的研究中出现的矩阵模型。这种移动边界现象通过癌症生长、晶体形成、冰融化、石油储量、一氧化碳封存和等离子体动力学来说明。PI由一群热情的博士生协助，他通过几个合作作品与数学，物理或工程等领域的专家联系。