Algebraic Structures on Polytopes

Explore algebraic structures on polytopes in this AMS Invited Address from the 2018 Joint Mathematics Meetings. Delve into combinatorics through a personal perspective, examining permutations and associations in polytopes. Discover motivating applications in inverting power series, including multiplication and composition. Learn about Hopf monoids, their definitions, and examples in graphs, posets, and matroids. Investigate the antipode of Hopf monoids and generalized permutahedra, including their coproduct and connections to posets, graphs, and matroids. Examine characters of Hopf monoids and their groups for permutahedra and associahedra. Conclude with current directions in polytope algebra, gaining insights from Federico Ardila of San Francisco State University in this comprehensive 58-minute lecture.

Intro
What is combinatorics about? A personal view
11. A tale of two polytopes Permutations
1.2. A tale of two polytopes: Associations
13. Motivating application Inverting power series
Inverting power series: Multiplication
Inverting power series Composition
21. Hopf monoids: Definition
The Hopf monoid of graphs.
The Hopf monoid of posets
The Hopf monoid of matroids
22. The antipode of a Hopf moncid
Examples: The antipode of a graph, matroid, poset
Generalized permutahedra in 3-D
3.3. The Hopf monoid GP Coproduct
The Hopf monoid of generalized permutahedra.
3.4. Generalized permutahedra: Posets, graphs, matroids
3.5. The antipode of GP
The antipodes of graphs, matroids, posets
Many antipode formulas
Characters of Hopf monoids.
The group of characters for permutahedra
The group of characters of associahedra
6.1. Current direction 1: the polytope algebra.