Math 5594H: Honors mathematics topics- Algebraic combinatorics

Fall 2004

Prerequisites: Math 3593H with grade of at least B, experience in writing proofs, or dept consent. No combinatorics background, nor abstract algebra is required.
However, a solid background in linear algebra will be assumed, e.g.
  • determinants
  • eigenvalues, eigenvectors,
  • diagonalization of real symmetric matrices, and the notion of positive definiteness
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
Classes: Mondays and Wednesdays, 3:35-5:00 PM, in Lind Hall 320
Office hours: Mondays and Wednesday 2:30-3:20pm, Thursdays 10:10-11am, and also by appointment.  
Course content: Algebraic combinatorics deals with combinatorial problems (e.g. counting problems) where a direct approach is cumbersome or impossible, but where a bit of algebra can be effective.
This course is based on one that was given to Harvard math undergrads by Richard Stanley (MIT), and contains a selection of such topics and techniques. Here are the chapter titles from his lecture notes, to give some flavor of the topics:
  • Walks in graphs
  • Cubes and the Radon transform
  • Random walks
  • The Sperner property (in partially ordered sets)
  • Group actions on Boolean algebras
  • Young diagrams and q-binomial coefficients
  • Enumeration under group action (Polya theory)
  • A glimpse of Young tableaux
  • The matrix-tree theorem
  • Eulerian digraphs and oriented trees
  • Cycles bonds and electrical networks
Text: Topics in algebraic combinatorics by Richard Stanley
These are the lecture notes from the course that he taught at Harvard in Fall 2000.
I will distribute paper copies to the students early in the course.
(There is no text to buy from a bookstore).
Homework: There will be homework assignments due every other week (except weeks with exams) at the beginning of the Wednesday class, starting with Wednesday Sept. 22. The assignments will mainly be from the problem sets at the end of Stanley's lecture notes. I will try to hand out brief solutions or solution outlines. Late homework will not be accepted. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page with whom they have collaborated.  
Exams and grading:  There will be one take-home midterm exam and one take-home final exam, given out on dates to be determined later. The midterm will be worth 20%, the final 25%, and the remaining 55% of the course grade will be based on the quality and quantity of homework turned in.
Both the take-home midterm and final exams are to be open-book, open-notes, but there is to be no collaboration; the only human source you will be allowed to consult is the instructor.  
Policy on incompletes:  Incompletes will be given only in exceptional circumstances, where the student has completed almost the entire course with a passing grade, but something unexpected happens to prevent completion of the course. Incompletes will never be made up by taking the course again later. You must talk to me before the final exam if you think an incomplete may be warranted.  
Other expectations  This is a 4-credit course, so I would guess that the average student should spend about 8 hours per week outside of class to get a decent grade. Part of this time each week would be well-spent making a first pass through the material in the book that we anticipate to cover in class that week, so that you can bring your questions/confusions to class and ask about them.
Homework assignments
Assignment or Exam Due date Problems
Homework 1 Wed Sept. 22 Stanley's Problem Set #1, except for Problem 5
Homework 2 Wed Oct. 6 Stanley's Problem Set #1, Problem 5
All of Stanley's Problem Set #2.
Homework 3 Wed Oct. 20 Stanley's Problem Set #3, Problems 1,2,3,4
Midterm Exam Wed Oct. 27 Handed out in class
Homework 4 Wed Nov. 17 Stanley's Problem Set #3, Problem 5
All of Stanley's Problem Set #4
Homework 5 Wed Dec. 1 All of Stanley's Problem Set #5
Homework 6 Wed Dec. 8 Stanley's Problem Set #6, Problems 1,2
Stanley's Problem Set #7, Problems 1,2
Final exam Wed Dec. 15 Stanley's Problem Set #6, Problem 3
Stanley's Problem Set #7, Problem 3
Stanley's Problem Set #8, Problems 2,4,5

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