Hopf monoids in Species and associated Hopf algebras

I plan to give an overview of joint work in progress with Swapneel Mahajan.
The first lecture will be on category theory, the second on species and Hopf algebras, the third on deformations and higher dimensional generalizations.
We study the tensor category of species and relate it to the tensor category of graded vector spaces by means of bilax tensor functors. A substantial theory of abstract bilax tensor functors is developed first and then applied in this context. Constructions of Stover of graded Hopf algebras from Hopf monoids in species are then derived from the general theory. Deformations and higher dimensional generalizations of these constructions are prompted by the categorical approach. We study several specific examples of Hopf monoids in species and the graded Hopf algebras that correspond to them under the bilax tensor functors. We use the geometry and combinatorics of the Coxeter complex of type A to construct Hopf monoids and understand their interconnections. The corresponding Hopf algebras include those of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, other Hopf algebras of prominence in the recent literature, and new ones. This relates to recent interesting work of Patras with Livernet, Reutenauer, and Schocker.