Spectral analysis of geometric shapes

该项目的目的是统一三个不同的矩阵模型(一个是量子的，另一个是随机/热力学的，第三个是纯代数的)，与平面椭圆生长有关，平面椭圆生长是一种界面动力学，涵盖了大量的自然和理论现象。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，以及最近关于这些主题的大量收敛的研究使得该项目及时且不可避免。作为数学工具，该提议将依赖和发展复正交多项式、矩矩阵和平面或空间形状研究中出现的潜在理论算子的新方面。将特别强调从间接测量中恢复阴影函数的最大熵方法，例如部分几何层析数据和随机非厄米矩阵频谱的渐近性。经过十年在流体力学和量子物理中的激烈和充满活力的发现，本提案解决了现代科学中这些特定领域高度感兴趣的一系列数学问题。将平面形状和体积几何编码为数字或代数符号的传统可以追溯到17世纪笛卡尔及其追随者在今天所称的解析几何中具有里程碑意义的贡献。很久以后，我们今天都受益的电、磁和原子科学的发展和应用，通过朝着同一方向迈出的第二个主要步骤成为可能，即将复杂的物理实体表示为称为矩阵的大型数字或符号阵列。建议的项目侧重于研究当前流体或更复杂但类似介质的运动界面的基质模型。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都说明了这种移动边界现象。PI由一群热情的博士生协助，通过几项合作工作，他与数学、物理或工程等领域的专家建立了联系。

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.