Math 7410, Topics in Combinatorics, Fall 2016
I. Hopf monoids in species
Aug 23. Species and enumeration. Cauchy and substitution products.
Aug 25. Monoids in species. Free monoids. Comonoids and bimonoids.

Aug 30. Monoidal categories. Monoids and comonoids. Convolution.
Sep 1. Braided monoidal categories. Bimonoids and Hopf monoids. Antipode.

Sep 6. Linearization. Hopf monoids in vector species. Antipode of L.
Sep 8. q-braiding. q-deformation of L. Connected bimonoids.

Sep 13. Antipode formulas. Examples.
Sep 15. Group of characters. Reciprocal of a power series.

Sep 20. Substitution of power series and Lagrange inversion.
Sep 22. Polynomial invariants. Chromatic polynomial and order polynomial.

Sep 27. Reciprocity. Antipode of E and L and the Coxeter complex.
Sep 29. Coxeter complex and Tits product. Higher Hopf monoid axioms. Antipode as Euler characteristic of a pair.

Oct 4. Normal fans of polytopes. Standard permutahedron and generalized permutahedra.
Oct 6. Hopf monoid of generalized permutahedra and related examples.

II. Hyperplane arrangements
Oct 11. No class (Fall break)
Oct 13. Real hyperplane arrangements. The braid arrangement. Faces and flats. Tits monoid and sign sequences.

Oct 18. Tits monoid and partial order on faces. Species and Hopf monoids relative to a hyperplane arrangement.
Oct 20. Hopf monoids as modules over the Janus algebra.

Oct 25. Review of Möbius functions. Weisner's formula. Algebra of a lattice. Semimodular lattices.
Oct 27. Zaslavsky's formula for chambers and faces. Restriction and contraction. Characters.

Nov 1. Primitive elements. Lie and Zie elements. Dimensions of Lie and Zie.
Nov 3. Generic hyperplanes. Dynkin idempotent. Zaslavsky for bounded chambers. Dynkin basis of Lie.

Nov 8. Braid arrangement, classical Dynkin. Global operads.
Nov 10. Joyal-Klyachko-Stanley.

Nov 15. Lunes. Lune category.

III. More on monoidal categories
Nov 17. Schur functors, Schur-Weyl duality. Fock functors.

Nov 22. Monoidal functors.
Nov 24. No class (Thanksgiving).

Nov 29. Simplicial objects and chain complex functor.
Dec 1. 2-monoidal categories. Eckmann-Hilton argument.