Math 824 (Algebraic Combinatorics)
Fall 2016
General Information |
Lecture Notes |
Problem Sets |
Textbooks |
Final Project |
Links
General Information
- Instructor:
Jeremy Martin
- E-mail:
(the best way to contact me)
- Office: 618 Snow Hall, (785) 864-7114
- Office hours: Wednesdays, 3-4pm, or by appointment
- Lectures:
MWF, 1:00-1:50pm, 456 Snow
- Prerequisite: Math 724 (Enumerative Combinatorics) or permission of the instructor. Math 790 (Linear Algebra) is not an official requirement but is strongly encouraged.
- KU course number: 66459
- Syllabus
Lecture Notes
Lecture
Notes on Algebraic Combinatorics (one big PDF file; last update
12/7/16)
These notes are licensed under a Creative
Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
In short, use them freely but do not sell them or anything derived from
them. If you are a KU graduate student (or for that matter, if you
aren't), do not print out the full set of notes on the department's
printer!
Lecture topics, day by day:
- Mon 8/22: Posets - basic definitions and examples
- Wed 8/22: Young's lattice; simplicial complexes
- Fri 8/24: Stanley-Reisner rings; f- and h-vectors; shellability
- Mon 8/29: Polytopes
- Wed 8/31: The fundamental theorem; the Bruggesser-Mani proof of the Dehn-Sommerville relations
- Fri 9/2: Distributive lattices
- Mon 9/5: No class (Labor Day)
- Wed 9/7: Proof of the FTFDL
- Fri 9/9: Modular lattices
- Mon 9/12: Semimodular lattices
- Wed 9/14: Geometric lattices
- Fri 9/16: Matroid closure operators and rank functions
- Mon 9/19: Graphic matroids; independence systems
- Wed 9/21: Equivalence of rank functions and independence systems; matroid complexes; basis systems
- Fri 9/23: Other matroid formulations; representability; the Fano and non-Pappus matroids
- Mon 9/26: Direct sum and duality
- Wed 9/28: Deletion and contraction
- Fri 9/30: The Tutte polynomial: examples and proof of equivalence of the closed-form and recursive formulas
- Mon 10/3: The Tutte polynomial: recipe formulas and activities
- Wed 10/5: The Tutte polynomial: colorings and orientations
- Fri 10/7: Guest lecture by Hailong Dao
- Mon 10/10: No class (Fall Break)
- Wed 10/12: The incidence algbra and the Möbius function
- Fri 10/14: Möbius inversion
- Mon 10/17: The Möbius algebra; Weisner's theorem
- Wed 10/19: The crosscut theorem; introduction to hyperplane arrangements
- Fri 10/21: Hyperplane arrangements: basic facts, examples, counting regions
- Mon 10/24: Zaslavsky's theorems
- Wed 10/26: The finite field method; supersolvable arrangements
- Fri 10/28: Guest lecture by Marge Bayer: Ehrhart theory
- Mon 10/31: Introduction to representations
- Wed 11/2: Proof of Maschke's theorem; characters and functors
- Fri 11/4: Schur's Lemma and the Fundamental Theorem of Character Theory
- Mon 11/7: Character tables; one-dimensional representations and Pontrjagin duality
- Wed 11/9: Representations of the symmetric group; restriction and induction (barely)
- Fri 11/11: Symmetric functions I (Marge Bayer)
- Mon 11/14: Symmetric functions II (Marge Bayer)
- Wed 11/16: Symmetric functions III (Marge Bayer)
- Fri 11/18: Schur functions: proof of symmetry
- Mon 11/21: The Cauchy and dual Cauchy kernels and the Hall inner product
- Wed 11/23: No class - Thanksgiving
- Fri 11/25: No class - Thanksgiving
- Mon 11/28: The Robinson-Schensted-Knuth correspondence; generalized RSK and Schur functions
- Wed 11/30: Restricted and induced representations and Frobenius reciprocity; the Frobenius characteristic map
- Fri 12/2: The hook-length formula
- Mon 12/5: Presentations (Smita and Amanda)
- Wed 12/7: The Jacobi-Trudi determinant definition of Schur functions
Problem Sets
Problem sets will be assigned approximately biweekly, at least one week before the
deadline.
All solutions must be typeset using LaTeX. E-mail me your solutions as a PDF file under the name
{your-last-name}{number-of-problem-set}.pdf (e.g., "Rota5.pdf"). In general, the
deadline is 11:59pm of the due date.
- Problem Set 1, due Tuesday 9/6
- Problem Set 2, due Friday 9/16
- Problem Set 3, due Monday 10/3: Exercises
3.1, 3.2, 3.6, 3.7, 3.8, and at least one of 3.9 and 3.10
- Problem Set 4, due Friday 10/21: Exercises 4.3, 4.4, 5.1, 5.2
- Problem Set 5, due Friday 11/11: Exercises 5.3, 5.6, 6.2, 6.3, 8.1, 8.2
- Problem Set 6, due Thursday 12/8: Exercises 8.3, 8.4, 9.2, 9.4, 9.5, and at least one of 8.6 or 9.3
Additional resources:
- Here is a header file with useful macros
(last update 8/26/16).
- Here is a sample source
file that uses the header (and will produce the first couple of pages of the lecture notes).
- Packages for making figures include TikZ, Ipe, and xfig.
Textbooks
We will follow the lecture
notes rather than any one specific textbook.
However, the following books may be
helpful (and you should definitely obtain the free downloads). All
these books can be perused in Jeremy's office.
- Richard P. Stanley, Enumerative Combinatorics, volume 1, 2nd ed. (Cambridge, 1997)
(Enumeration; posets and lattices; generating functions)
Buy it from the publisher or
download the free preprint version from Stanley's website
- Richard P. Stanley, Enumerative Combinatorics, volume 2 (Cambridge, 1999)
(More enumeration, including exponential generating functions; symmetric functions)
Buy it from the publisher
- Martin Aigner, A Course in Enumeration (Springer, 2007)
(Enumeration; posets, lattices, and matroids)
- Martin Aigner, Combinatorial Theory (Springer, 1997)
(Enumerative combinatorics, symmetric functions, and matroids)
- Richard P. Stanley, Hyperplane Arrangements
(lecture notes available free online)
- Alexander Schrijver, A Course in Combinatorial
Optimization (lecture notes available free online)
- Thomas Brylawski and James Oxley, The Tutte polynomial and its
applications, Chapter 6 of Matroid applications, N. White, ed.
(Cambridge Univ. Press, 1992)
- Matthias Beck and Raman Sanyal, Combinatorial Reciprocity
Theorems: A Snapshot of Enumerative Combinatorics from a Geometric
Viewpoint (manuscript available free online)
- Bruce
Sagan, The Symmetric Group, 2nd edn. (Springer, 2001)
- Federico
Ardila, Algebraic and geometric methods in enumerative
combinatorics (free book; covers much of the same material as my
notes but from a slightly different perspective, and probably a more
polished presentation)
Final Project
- The assignment. Everyone should meet with Jeremy and choose their paper by October 28.
- The schedule of talks and reviewers is below. If the links to papers don't work, try searching for the paper using MathSciNet
(requires KUID).
Date |
Presenter |
Reviewer |
Paper |
December 5 |
Smita Praharaj |
Amanda Wilkens |
A. Knutson and T. Tao
Honeycombs and sums of Hermitian matrices
Notices Amer. Math. Soc. 48, no. 2 (2001), 175-186. |
December 5 |
Amanda Wilkens |
Smita Praharaj |
B.E. Sagan
Probabilistic proofs of hook length formulas involving trees
Sém. Lothar. Combin. 61A (2009/10), Art. B61Ab, 10 pp.
|
December 14 |
Brandon Caudell |
Connor Smith |
J. Edmonds
Minimum partition of a matroid into independent subsets
J. Res. Nat. Bur. Standards Sect. B 69B (1965), 67-72.
|
December 14 |
Joseph Doolittle |
Brandon Caudell |
R.P. Stanley
Two poset polytopes
Discrete Comput. Geom. 1 (1986), 9-23. |
December 14 |
Arturo Jaramillo |
Bibekananda Mishra |
A. Nica and R. Speicher
Non-crossing partitions and free cumulants
From Lectures on the combinatorics of free probability (Cambridge U. Press, 2006), pp. 135-194 (mainly pp. 173-194) |
December 14 |
Bibekananda Mishra |
Joseph Doolittle |
P. Orlik and L. Solomon
Combinatorics and topology of complements of hyperplanes
Invent. Math. 56 (1980), no. 2, 167-189. |
December 14 |
Connor Smith |
Arturo Jaramillo |
A. Goodall
Fourier analysis on finite Abelian groups: some graphical applications
From Combinatorics, Complexity and Chance: a tribute to Dominic Welsh (G. Grimmett and C. McDiarmid, eds.), Oxford U. Press, 2007, 103-129. |
Links
KU links
Software and online resources
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Last updated Fri 12/9/16 8:00am