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.
