| Date | Chapter | Description | |
|---|---|---|---|
| Sep 1 | Simplicial complexes: CW-structure, examples | ||
| Sep 3 | Simplicial complex examples, Stanley-Reisner rings | ||
| Sep 6 | Labor Day! (no class) | ||
| Sep 8 | The Nerve Lemma and Helly Theorem | ||
| Sep 10 | Alexander Duality; proof of Topological Helly Theorem | ||
| Sep 13 | Poset topology and Möbius inversion | ||
| Sep 15 | Hall's Theorem: Möbius = reduced Euler | ||
| Sep 17 | Lattices; Hall's application of Möbius inversion | ||
| Sep 20 | Poset duality, complements | ||
| Sep 22 | Crapo and Homotopy Complementation Theorems | ||
| Sep 24 | Homotopy Complementation proof | ||
| Sep 27 | Homotopy Complementation applications | ||
| Sep 29 | Partition lattices, distributive and modular lattices | ||
| Oct 1 | Modular and left-modular elements | ||
| Oct 4 | Homotopy of lattices with modular chains via complementation | ||
| Oct 6 | EL-labelings | ||
| Oct 8 | Left-modular elements and labelings | ||
| Oct 11 | Supersolvable lattices | ||
| Oct 13 | Semimodular lattices | ||
| Oct 15 | Fall Break! (no class) | ||
| Oct 18 | Semimodular lattice examples, geometric lattices | ||
| Oct 20 | Matroids <-> geometric lattices | ||
| Oct 22 | EL-labelings of semimodular lattices | ||
| Oct 25 | Shellings, rearrangement lemmas | ||
| Oct 27 | Collapsing, homotopy type of shellable complexes | ||
| Oct 29 | EL-labelings give shellings | ||
| Nov 1 | Cohen-Macaulay complexes and local homology | ||
| Nov 3 | The dunce cap; various definitions of skeleton | ||
| Nov 5 | sequentially Cohen-Macaulay complexes | ||
| Nov 8 | connectivity parameter: depth | ||
| Nov 10 | depth is a topological invariant; Krull dimension | ||
| Nov 12 | (ring theoretic) depth | ||
| Nov 15 | Local cohomology of a ring: Koszul and modified Cech complexes | ||
| Nov 17 | Cohomology <-> Local cohomology of Stanley-Reisner | ||
| Nov 19 | Hilbert series and h-vectors | ||
| Nov 22 | h-vectors, M-vectors, and the Upper Bound Theorem | ||
| Nov 24 | Thanksgiving! (no class) | ||
| Nov 26 | Thanksgiving! (no class) | ||
| Nov 29 | h-vectors of shellable and partitionable complexes | ||
| Dec 1 | CL-labelings; vertex-decomposability | ||
| Dec 3 | Discrete Morse theory: connection with continuous Morse theory | ||
| Dec 6 | Discrete Morse theory: Morse functions give collapsing | ||
| Dec 8 | Discrete Morse theory: examples, Morse matchings | ||
| Dec 10 | Discrete Morse theory: shellings, brief idea of poset Morse theory | ||