Algebra and Combinatorics of Free Structures

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This project will deal with a variety of algebraic objects that havearisen recently in different areas such as Hopf algebra theory,combinatorial representation theory,the theory of operads, free probability theory, and renormalization theoryin 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 roleplayed by these objects in the other theories.Freeness is also responsible for the rich combinatorics exhibited by theseobjects. 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 foundapplications in free probability theory, while others are at the basis ofrenormalization theory. All of them are closely related to symmetric functionsand the representation theory of the symmetric group. A common feature is theexistence of Hopf algebraic structures, which are often also free in one senseor 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 interestboth 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 combinatoricsunderlying various free algebraic structures.The PI will introduce graduate students to this area of research and makeinternational contacts and collaborations in Europe, Latin America, and India.