Math 287Z - Geometric and Topological Combinatorics - Spring 287Z

Lectures: Monday and Wednesday 9:00am-10:15am, Science Center 507.

Lecturer: Lauren Williams (office Science Center 510 , e-mail williams@math.harvard.edu)

Office Hours: TBD

Course Assistant: Jiyang (Johnny) Gao

Course description

This class will be an introduction to topological and geometric combinatorics at the graduate level, covering several general areas:
(1) Posets
(2) Simplicial complexes
(3) Matroids
(4) Polytopes

One of the main themes of the class will be the question ``To what extent do combinatorial properties of an object determine its topology or geometry?" For example, to what extent does the face lattice of a simplicial or cell complex determine its homotopy type? To what extent does the graph of a polytope determine the polytope? And what kinds of combinatorial techniques can we use to then understand the topology of the object in question? Along the way we will discuss interesting examples coming from posets, polytopes, matroids, Coxeter groups, the Grassmannian and flag varieties, etc.

Course Prerequisites

I will assume that student have a strong background in algebra (familiarity e.g. with commutative rings). It will also be helpful to have some exposure to algebraic topology, including topics such as homology and homotopy.

Textbooks

Recommended reading: Enumerative Combinatorics I, 2nd edition (Richard Stanley), Matroid Theory (James Oxley), Lectures on Polytopes (Gunter Ziegler).

Grading

Grading will be based on homework (50%), and a presentation (10%) and final paper (40%) on a topic of your choice. If you work with others on the homework, you must write up your solutions independently, and list the names of the students you worked with. The final paper is due on May 1st, 2024.

Lectures

The following is a first approximation of the schedule for the course.

  • Lecture 1 (Jan 22): Introduction of the main characters: simplicial complexes, posets, matroids, polytopes.
  • Lecture 2 (Jan 24): Poset terminology, lattices, linear extensions.
  • Lecture 3 (Jan 29): The incidence algebra, Mobius inversion, and the Euler characteristic of the order complex.
  • Lecture 4 (Jan 31): Shellability. A shellable simplicial complex is homotopy equivalent to a wedge of spheres. EL labelings.
  • Lecture 5 (Feb 5): An EL labeling induces a shelling of the order complex. EL labelings for distributive lattices.
  • Lecture 6 (Feb 7): CW complexes and their face posets.
  • Lecture 7 (Feb 12): Bruhat order and shellability.
  • Lecture 8 (Feb 14): Discrete Morse theory.
  • Lecture 9 (Feb 21): Discrete Morse functions as matchings. Start on Grassmannians.
  • Lecture 10 (Feb 26): Grassmannians and flag varieties, Schubert cells.
  • Lecture 11 (Feb 28): Matroids (axiom systems from independence, bases).
  • Lecture 12 (March 4): The Gelfand-Goreksy-MacPherson-Serganova theorem on matroid polytopes.
  • Lecture 13 (March 6): Applications of the GGMS theorem. Operations on matroids and the Tutte polynomial.
  • Lecture 14 (March 18): The Tutte polynomial.
  • Lecture 15 (March 20): Log-concavity and Lorentzian polynomials.
  • Lecture 16 (March 25): The matroid stratification. Positroids and positroid polytopes.
  • Lecture 17 (March 27): Oriented matroids and the matroid Grassmannian. Positively oriented matroids.
  • Lecture 18 (April 1): The face lattice of polytopes. The cyclic polytope; the permutohedron.
  • Lecture 19 (April 3): Face numbers of polytopes. H-vectors of simplicial polytopes via shelling.
  • Lecture 20 (April 8): Simple polytopes and their graphs; graph-associahedra.
  • Lecture 21 (April 10): The g-theorem (which characterizes face numbers of simplicial polytopes) and generalizations.
  • Lecture 22 (April 15): Katherine and Sebastian (Hodge theory of matroids); Ziyong (Shellability of Bruhat order).
  • Lecture 23 (April 17): Lauren and Jennifer (Lattice structure for orientations of graphs); Wittmann (Discrete Morse theory).
  • Lecture 24 (April 22): Eric (Miracle of integer eigenvalues); Jacob (Two poset polytopes); Ilaria (F-polynomials from posets).
  • Lecture 25 (April 24): Nadine and Dora (Permutohedra, associahedra, beyond); Gregory (Bruhat interval polytopes).
  • FINAL PROJECT DUE (May 1st)