RUI: Operator Theory and Its Connections

这个提议围绕着两个新颖的线索，在过去的几年里，这两个线索都已经为PI，他的学生和他的合作者带来了丰硕的成果。第一个涉及算子理论（最初是为量子力学提供严格的基础），组合学（经常处理离散枚举问题的领域）和概率论（研究机会和随机性）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将招募各种各样的研究生来解决这些问题。作为一名获得无数教学奖项的获奖教师，PI能够激励广大学生攻读数学科学学位。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（组合数学中的主要对象）中因子分解长度的渐近问题。某些对称多元多项式的新概率实现在这项工作中证明是至关重要的，并提供了经典的正性结果的证明，同时揭示了广泛的推广。反过来，这些结果承诺在算子理论和算子代数的新的应用。例如，他们的组合结果表明，统计处理某些家庭的AF代数。PI还寻求物理和光谱解释的特殊概率措施时出现的考虑某些组合过程的限制。这些措施是独立的利益，从计算机辅助设计（样条理论）的结果已经照亮了他们的一些属性。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富有成效的。例如，结果的PI和他的合作者对渐近的Toeplitz矩阵所产生的问题的启发格理论。在另一个方向，框架理论的结构和运营商相关的方法已被用来证明新的结果lattice.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.