This project will deal with a variety of algebraic objects that have arisen recently in different areas such as Hopf algebra theory, combinatorial representation theory, the theory of operads, free probability theory, and renormalization theory in mathematical physics. It should be regarded as an extension of the classical work on the theory of free Lie algebras and its interplay with the representation theory of the symmetric group. Through the eyes of operad theory, the objects we deal with are free algebras of various kinds. Addressing this property is fundamental in understanding the role played by these objects in the other theories. Freeness is also responsible for the rich combinatorics exhibited by these objects. Part of the goal of the project is to make the algebraic structure as explicit as possible, which often leads to interesting combinatorial constructions. Conversely, the project will make use of algebraic properties to unify and generalize important constructions in combinatorics. Some of these free algebras have found applications in free probability theory, while others are at the basis of renormalization theory. All of them are closely related to symmetric functions and the representation theory of the symmetric group. A common feature is the existence of Hopf algebraic structures, which are often also free in one sense or another. Exploiting this structure is another central feature of the project.
Combinatorics provides a tool for handling subtle algebraic structures. Understanding these structures is important in various areas of recent interest both in pure mathematics (free probability theory, representation theory) as well as in mathematical physics (renormalization theory). This project plans to deepen our understanding of the rich combinatorics underlying various free algebraic structures. The PI will introduce graduate students to this area of research and make international contacts and collaborations in Europe, Latin America, and India.
