Rota-Baxter Operators in Physics and Mathematics

本计画研究Rota-Baxter算子及数学物理、代数、数论等相关运算。这项工作探讨了Rota-Baxter代数在微扰量子场论的重整化过程和其他黎曼-希尔伯特对应实例中的应用。该项目还将研究Rota-Baxter代数与量子理论、组合学和微分方程数值解中的分解之间的联系。这个项目的代数方面考虑了Rota-Baxter算子与另外两个代数的联系，这两个代数的乘积是由shuffle积的推广定义的，并且是两个二进制运算的和。这三种代数在物理学、数论和组合学中发挥着重要作用，它们之间的密切联系将在更广泛的背景下加以考虑。这项研究旨在为这些更一般的代数结构及其应用的理论理解做出重大贡献。该项目还将探讨Rota-Baxter算子和相关算子在研究多个zeta值及其在数论中的推广中所起的基本作用。Rota-Baxter算子是分析中积分算子的推广。这个算子的研究是由数学家独立进行的，他们的动机来自概率论，而物理学家的研究则与经典的杨-巴克斯特方程的解有关。随后在物理和数学的几个领域发现了进一步的重要应用。最引人注目的是算子与最近量子场论的代数方法的关系，它既澄清了量子场论的理论基础，又大大简化了其计算。Rota-Baxter算子的高度递归、树状定义也引起了人们对其在计算数学和组合学中的应用的兴趣。这个项目研究了在数学物理、代数和数论中感兴趣的算子的性质和推广。

This project studies the Rota-Baxter operator and related operators of interest in mathematical physics, algebra, and number theory. The work explores applications of Rota-Baxter algebras to the renormalization process of perturbative quantum field theory and to other instances of the Riemann-Hilbert correspondence. The project will also pursue connections of Rota-Baxter algebras with decompositions in quantum theory, combinatorics, and numerical solutions of differential equations. The algebraic aspect of this project considers the Rota-Baxter operator in connection with two other algebras whose products are defined by a generalization of the shuffle product, and as a sum of two binary operations. The close connections among these three algebras, which play prominent roles in physics, number theory, and combinatorics, will be considered in broader contexts. This investigation aims to contribute significantly to the theoretical understanding of these more general algebraic structures and their applications. The project will also explore the fundamental role played by Rota-Baxter operators and related operators in the study of multiple zeta values and their generalizations in number theory.The Rota-Baxter operator is a generalization of the integration operator in analysis. The study of this operator was independently carried out by mathematicians, with motivation from probability theory, and by physicists, in connection with solutions of the classical Yang-Baxter equation. Further important applications were subsequently found in several areas of physics and mathematics. Most striking is the relation of the operator to a recent algebraic approach to quantum field theory, which both clarifies the theoretical foundation of quantum field theory and substantially simplifies its calculations. The highly recursive, tree-like definition of the Rota-Baxter operator has also generated interest in its application to computational mathematics and combinatorics. This project studies the properties of the operator and generalizations of interest in mathematical physics, algebra, and number theory.