Title: Rota-Baxter Algebra and Related Topics

Abstract:
  

The Rota-Baxter algebra is an abstraction of the algebra of continuous functions acted by the integral operator. It originated with the probability study of Glenn Baxter in 1960 and was developed further by Cartier and the school of Rota in the 1960s and 1970s. Independently, this structure appeared in the Lie algebra context as the operator form of the classical Yang-Baxter equation started in the 1980s. Since the 1990s,Rota-Baxter algebra has found further  heoretical developments and found applications in number theory, combinatorics and mathematical physics.

  

This course will start with a minimum background on algebra and will cover the basic theory of Rota-Baxter algebra and its applications. We will discuss three aspects of Rota-Baxter algebras: its operator aspect that emphasize the properties of the Rota-Baxter operator and its applications to Waring's identity of symmetric functions and to Connes-Kreimer's Hopf algebra approach of quantum field theory renormalization; its product aspect that studies the algebraic structure of Rota-Baxter algebras and its relationship with multiple zeta values; and its operad  aspect that establishes the connection between Rota-Baxter algebra and the operads of dendriform dialgebra and trialgebra, especially the Hopf algebras of rooted trees of Loday and Ronco.