Combinatorial Hopf Algebras

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This project will contribute to establish Hopf algebraic combinatorics as a major component within the wider context of algebraic combinatorics, by developing new results that unify and generalize important constructions in combinatorics. The same Hopf algebras that are at the basis of these constructions also have deep connections with other areas of mathematics and physics such as the theory of operads, renormalization theory, and representation theory. Thus, this project will also deal with abstract constructions in the theory of associativity breaking and the corresponding types of algebras, the investigation of the Hopf algebras of trees of quantum electrodynamics, and the investigation of various objects closely related to the algebra of symmetric functions, such as quasi-symmetric and non-commutative symmetric functions.Combinatorial questions study various ways of counting. In this proposal methods from algebra will be used to about new insight into some of these counting questions. In addition the PI will introduce graduate students to this area of research and make international contacts and collaborations in Latin America and Europe.