ROTA-BAXTER ALGEBRA AND RELATED STRUCTURES

This project continues the study of the application and theory of Rota-Baxter algebra and related structures that have arisen from the interaction between mathematics and physics. The application part of this project pursues the mathematical understanding of the renormalization of perturbative quanturm field theory and the application of the renormalization method in the study of mathematical objects, such as divergent multiple zeta values and Riemann integrals. Multivariable special values of convex cones and their renormalization will also be studied in connection with the Todd class of the related toric varieties. A relative generalization of the Rota-Baxter operator called the O-operator will be studied in the context of Yang-Baxter equations and integrable systems. The theoretical part of the project considers the classification, representation and chain conditions of these structures, and the relationship among Rota-Baxter related operators, shuffle type products and special operads such as the dendriform, pre-Lie and PostLie operads. The Rota-Baxter operator is an algebraic 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, as solutions of the classical Yang-Baxter equation. Further important applications were subsequently found in several areas of physics and mathematics, including quantum field theory, Yang-Baxter equations, number theory, operads and Hopf algebras. The highly recursive, tree-like definition of the Rota-Baxter operator has also generated interest in its connection with computational mathematics and combinatorics. Further study of this operator will contribute to the understanding of the role it plays in such diverse areas.