Math 580A: Topics in Combinatorial Analysis
Lattice Points and Biased Graphs
Fall 2005

This course has two parts that are related, although it is not immediately obvious. The theme, however, is to introduce you to research topics in combinatorics in which I am interested. I intend to present the topics in a way that makes them accessible without requiring you to have a combinatorics background. There will be a lot of geometry.

The course is not an introductory graduate course. The absolute minimum requirement is a good understanding of abstraction and a solid modern algebra background (as from a graduate course), and the more graduate math you know, the better (that's the famous "mathematical maturity"). If you aren't sure whether you might be interested or ready for this class, please see me.

We meet on MWF 1:10 - 2:10 in LN-2205.
Special class times:
Week of October 3: No class.
Week of November 7: No classes.
We will make up those four missed classes at other times, which will be arranged so everyone can be there.
Special class times:
Wed., October 10: 1:10 - 2:10
Thurs., November 3: 1:15 - 2:15 in LN-2201
Thurs., December 8: 1:15 - 2:30 in LN-2201

There is a Combinatorics Seminar that meets usually on Tuesdays at 4:15, but occasionally at a different time; check the schedule. All 580 students are expected to attend (if the room is big enough for 580 students).

Course Work

I will expect you, the students, to study the material and to work on as many of the exercises as you can. I will meet separately with each student frequently (every week, I hope) to discuss your progress and any questions you or I may have. I will frequently collect written work: see the homework assignments below and the lists of additional homework problems:

Course Outline

Short Outline

Class Schedule

Week ofTopicReadingHomework to hand in

Aug. 29-Sept. 2 Introductory survey.
Examples of lattice point counting.
Ch. 2

Sept. 7-9 Examples of lattice point counting. Ch. 2

Sept. 12-16 Lattice point counting.
Pick's, rational triangles.
Frobenius and geometry.
Ch. 2

Ch. 1
9/16: Three approved problems from Ch. 2.
Approved so far: 10, 15, 21, 22, 30 (if not from Thm. 2.9).

Sept. 19-23 Integral and rational
polytopes and cones.
Sect. 3.1-3 9/23: Three problems from Ch. 1.

Sept. 26-30 Integral and rational
polytopes and cones.
Sect. 3.3-5 9/26: Three problems from Ch. 3.
9/30: Presentations.

Oct. 3-7 (No class.)

Oct. 10-14 Curve fitting (interpolation). Sect. 3.6 10/12: Six more problems from Ch. 3.
10/10-12: Presentations.

Oct. 17-21 Rational polytopes and cones.
Sect. 3.7-8
Sect. 4.1-4
Sect. 5.1
10/21: Three problems from Ch. 4.

Oct. 24-28 Face numbers of simple polytopes.
Magic and semimagic squares (weak form).
Sect. 5.1-5.4
Sect. 6.1-6.3
11/2: Two problems from Ch. 5.

Oct. 31-Nov. 4 Magic and semimagic squares (weak form).
Hyperplane arrangements, intersection poset (matroid), lattice points.
Magic and semimagic squares.
Sect. 6.3-4
"IOP" and "MML"
11/16: Three problems from Ch. 6.

Nov. 7-11 Hyperplane arrangements and lattice points.
Magic and semimagic squares.
"IOP" and "MML" Student group discussions at regular class time.

Nov. 14-18 Hyperplane arrangements and lattice points.
Magic and semimagic squares.
Graphs: coloring, orientation, geometry.
"MML", "SLS"
11/21: Three problems from the "Hyperplanes" problem list.

Nov. 21-23 Graphs.
Signed graphs: coloring, geometry.

Nov. 28 - Dec. 2 Signed graphs: coloring, orientation, geometry. Three problems from the "Signed, Gain, and Biased Graphs" problem list.

Dec. 5-9 Gain graphs: dependence structures, geometry, coloring.
Biased graphs.
"BG1-4" ... especially "BG1" Sect. 5, "BG2" Sects. 2, 3, "BG3" Sect. 4, "BG4" Sects. 2.1, 4.1.

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