# Overview

We meet 10:00 to 11:00 on Mondays in Chase 107 to discuss simplicial complexes, matroids, hilbert functions, Stanley-Reisner rings, etc.

# Lecture Notes

Winter 2009:- Lecture 1 (Feb 11) - Jason Brown introduces simplicial complexes and matroids.
- Lecture 2 (Feb 18) - Jason Brown talks about Sperner and Kruskal-Katona bounds and operations on simplicial complexes.
- Lecture 3 (Mar 4) - Jason Brown talks about recursively computing $f$-vectors and certain classes of complexes.
- Lecture 4 (Mar 18) - Jason Brown talks about interval partitions, $h$-vectors and new formulas for reliability.
- Lectures 5 & 6 (Mar 25 & Apr 1) - Jason Brown shows Stanley's bound on the $h$-vector and relates $f$-vectors to quotient rings.
- Algebra Notes (Not Given) - Andrew's notes on Stanley-Reisner rings. This has not been presented, but someone *could* talk about it while I'm gone.

- Lecture 7 (Jan 18) - Andrew Hoefel talks about resolutions of monomial ideals, regularity and Hilbert polynomials.
- Lecture 8 (Feb 1) - Andrew Hoefel talks about the Taylor complex, linear strand, and some known Betti numbers.
- Lecture 9 (Feb 8) - Andrew Hoefel introduces simplicial homology.
- Lecture 10 (Feb 15) - Andrew Hoefel computes $\beta_{i,2i}$ using lcm-lattices, Hochster's formula, and barycentric subdivision. I'll be using notes from lectures 7,8,9.
- Lecture 11 (March 8) - Andrew Hoefel introduces homotopy and cell complexes.
- Lecture 12 (March 15) - Visitor Gwyn Whieldon talks about Discrete Morse Theory (part 1).
- Lecture 13 (March 22) - Visitor Gwyn Whieldon talks about Discrete Morse Theory (part 2).