Week 1: August 28 - September 2
Monday:
Walks in graphs.
(
lecture,
tex file)
Some links that might help with eigenvectors and eigenvalues:
Wiki page
for eigenvalues and eigenvectors,
Wiki page for
diagonalization,
see lectures 9, 10, and 11 in my
201
notes.
Wednesday:
Walks on the complete graph.
(
lecture,
tex file)
Friday:
Random walks.
(
lecture,
tex file)
Week 2: September 5 - 9
Monday:
No class today (Labor Day).
Wednesday:
Posets, chains, antichains, Sperner property.
(lecture, tex, slides)
(
lecture,
tex file)
Friday:
Order matchings from order-raising operators.
(
lecture,
tex file)
- Reading: Chapter 4 through the proof of Lemma 4.5.
Week 3: September 12 - 16
Monday:
\(B_n\) has the Sperner property.
(
lecture,
tex file)
- Reading: finish Chapter 4.
Wednesday:
The symmetric group. Group actions.
(
lecture)
Friday:
The quotient poset \(B_n/G\).
(
lecture)
- Reading: Chapter 5 through Theorem 5.8.
Week 4: September 19 - 23
Monday:
Review homework. Sperner property for \(B_n/H\).
(
lecture)
Wednesday:
Sperner property for \(B_n/H\). Young diagrams.
(lecture, tex file)
(
lecture)
Friday:
Log-concave sequences and polynomials with real zeros.
(lecture, tex file)
(
lecture,
tex file)
- Reading: Section of polynomials with real zeros at the end of
Chapter 5.
Week 5: September 26 - 30
Monday:
\(q\)-binomial coefficients and Young diagrams. Enumeration under symmetry.
(
lecture)
- Reading: Chapter 6 and beginning of Chapter 7.
Wednesday:
Enumeration under symmetry. Polya counting.
(
lecture)
Friday:
Polya counting.
(
lecture)
Week 6: October 3 - 7
Monday:
Proof of Polya's theorem.
(
lecture)
Wednesday:
Generating functions: first example; the ring of formal power series.
(
lecture)
Friday:
Calculus of ordinary generating functions.
(
lecture)
Week 7: October 10 - 14
Monday:
Exponential generating functions.
(
lecture)
Wednesday:
Applications.
(
lecture)
Friday:
Dirichlet series.
(
lecture)
Week 8: October 24 - 28
Monday:
Poset incidence algebra.
(
lecture)
Wednesday:
Möbius inversion for posets
(
lecture,
tex file)
Friday:
Applications of Möbius inversion
(
lecture,
tex file)
Week 9: October 31 - November 4
Monday:
Simplicial complexes.
(
lecture,
tex file)
Wednesday:
Simplicial complexes.
(
lecture,
tex file)
Friday:
Order complexes.
(
lecture,
tex file)
Week 10: November 7 - 11
Monday:
Cycles and cuts in graphs.
(
lecture)
- Reading: Reading: Chapter 11 of our text and the lecture notes.
Wednesday:
The discrete Laplacian. (lecture, tex file)
(
lecture,
tex file)
Friday:
Quick introduction to Smith normal form. (lecture, tex file)
(
lecture,
tex file,
slides).
Week 11 November: 14 - 18
Monday:
Structure theorem for finitely generated abelian groups.
(
lecture,
tex file,
slides).
Wednesday:
Matrix-tree theorem.
(
lecture,
tex file,
slides).
Friday:
A tree bijection.
(
lecture,
tex file)
Week 12: November 21 - 25
Monday:
Consequences of the matrix-tree theorem;
tree bijections and kappa-inversions; rotor router example.
Rotor routing and abelian
sandpile reference.
(
Mathfest presentation,
and see the lecture notes for last Friday.)
- Reading: lecture notes from last week.
Wednesday:
Firing posets I.
(
lecture,
tex file,
slides).
Friday:
Thanksgiving break. No class today.
Week 13: November 28 - December 2
Monday:
Distributive lattices.
(
lecture,
tex file,
slides).
Wednesday:
Firing posets II.
(
lecture,
tex file,
slides).
Friday:
Firing posets III.
(
lecture,
tex file)
Week 14: December 5 - 10
Monday:
Firing posets IV.
(
lecture)
Wednesday:
TBA (lecture, tex file)
Friday:
No class today.
The \(\LaTeX\) document preparation system