Math 535 - Schedule















Schedule of topics: Math 535

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

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