Date  Chapter  Description 

Sep 1   Simplicial complexes: CWstructure, examples 
Sep 3   Simplicial complex examples, StanleyReisner 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 leftmodular elements 

Oct 4   Homotopy of lattices with modular chains via complementation 
Oct 6   ELlabelings 
Oct 8   Leftmodular 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   ELlabelings of semimodular lattices 

Oct 25   Shellings, rearrangement lemmas 
Oct 27   Collapsing, homotopy type of shellable complexes 
Oct 29   ELlabelings give shellings 

Nov 1   CohenMacaulay complexes and local homology 
Nov 3   The dunce cap; various definitions of skeleton 
Nov 5   sequentially CohenMacaulay 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 StanleyReisner 
Nov 19   Hilbert series and hvectors 

Nov 22   hvectors, Mvectors, and the Upper Bound Theorem 
Nov 24   Thanksgiving! (no class) 
Nov 26   Thanksgiving! (no class) 

Nov 29   hvectors of shellable and partitionable complexes 
Dec 1   CLlabelings; vertexdecomposability 
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 