RUI: Operator Theory and Its Connections

这一提议围绕着两条新的线索，在过去的几年里，这两条线索都已经为少年派、他的学生和他的合作者结出了丰硕的果实。第一个涉及算符理论(最初是为量子力学提供严格的基础)、组合学(一个经常处理离散计数问题的领域)和概率论(研究偶然性和随机性的研究)之间的动态相互作用。第二条线索涉及离散几何(研究某些高度结构化的几何对象)和算子理论之间的相互作用。许多子项目适合本科生参与。PI将招募一系列不同的研究型学生来解决这些问题。作为一名获得过多个教学奖项的获奖教师，PI能够激励广泛的学生攻读数学科学学位。这个项目将产生新的数学和新的数学家。在最近的工作中，PI和他的合作者使用了算子理论、调和分析和随机矩阵理论中熟悉的分析工具来回答几乎所有关于数值半群(组合学中的主要对象)中因式分解长度的渐近问题。在这项工作中，证明了某些对称多元多项式的新的概率实现是至关重要的，并且还提供了经典正性结果的证明，同时揭示了更广泛的推广。反过来，这些结果有望在算子理论和算子代数中得到新的应用。例如，他们的组合结果建议对某些AF代数族进行统计处理。当人们考虑某些组合过程的极限时，PI还寻求对特殊概率度量的物理和光谱解释。这些测量是独立的，因为计算机辅助设计(样条法理论)的结果已经阐明了它们的一些性质。离散几何(格)和算子理论(Toeplitz行列式和框架)之间的联系也被证明是卓有成效的。例如，PI和他的合作者关于Toeplitz矩阵的渐近性的结果源于格子理论的启发。另一方面，框架理论的构建和运算符相关的方法已经被用来证明关于网格的新结果。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This proposal revolves around two novel threads, both of which have already borne copious fruit for the PI, his students, and his collaborators in the last few years. The first concerns the dynamic interaction between operator theory (originally developed to provide a rigorous foundation for quantum mechanics), combinatorics (a field that often treats discrete enumeration problems), and probability theory (the study of chance and randomness). The second thread concerns the interplay between discrete geometry (the study of certain highly structured geometric objects) and operator theory. Many of the sub-projects are suitable for undergraduate involvement. The PI will recruit a diverse array of research students to tackle these problems. As a decorated instructor who has earned numerous teaching awards, the PI is able to inspire a broad range of students to pursue degrees in the mathematical sciences. This project will produce new mathematics and new mathematicians.In recent work, the PI and his collaborators used analytic tools familiar in operator theory, harmonic analysis, and random matrix theory to answer virtually all asymptotic questions about factorization lengths in numerical semigroups (staple objects in combinatorics). New probabilistic realizations of certain symmetric multivariate polynomials proved crucial in this work, and also provided proofs of classical positivity results while revealing broad generalizations. In turn, these results promise new applications in operator theory and operator algebras. For example, their combinatorial results suggest a statistical treatment of certain families of AF algebras. The PI also seeks physical and spectral interpretations of the peculiar probability measures that arise when one considers limits of certain combinatorial processes. These measures are of independent interest, for results from computer-aided design (spline theory) have already illuminated some of their properties. The connection between discrete geometry (lattices) and operator theory (Toeplitz determinants and frames) has also proven fruitful. For example, results of the PI and his collaborators on the asymptotics of Toeplitz matrices arose from problems inspired by lattice theory. In the other direction, frame-theoretic constructions and operator-related methods have been used to prove new results about lattices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.