MATH 890.3

Seminar in Polytopes

Fall 2006

Meeting times: MWF 1:10-2:00, HH 439
Prerequisites: Advanced Linear Algebra (it's ok to take MATH 725 concurrently) or consent of the instructor

Instructor: Dr. Matthias Beck
Office: Thornton Hall 933
Office hours:
M 2:10-3:00
W 12:10-1:00
F 11:10-12:00
& by appointment
Phone: +1 415 405 3473

Polytopes are the natural generalizations of line segments and polygons to higher dimensions. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all flat. One way to define a polytope is to consider the convex hull of a finite collection of points in Euclidean space Rd. That is, suppose someone gives us a set of points v1, ..., vn in Rd. The polytope determined by the given points vj is defined by all linear combinations c1v1 + c2v2 + ... + cnvn, where the coefficients cj are nonnegative real numbers that satisfy the relation c1 + c2 + ... + cn = 1. This construction is called the vertex description of the polytope.

There is another equivalent definition, called the hyperplane description of the polytope. Namely, if someone hands us the linear inequalities that define a finite collection of half-spaces in Rd, we can define the associated polytope as the simultaneous intersection of the half-spaces defined by the given inequalities.

It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that flows naturally from their geometry.

Our goal in this seminar is to understand the combinatorial structure of polytopes. We will start by proving the equivalence of the two descriptions of polytopes given above and then proceed to think about questions such as 'which polytopes have the maximal number of faces given a fixed dimension and number of vertices (extreme points)?'

Text book: Günter M. Ziegler, Lectures on Polytopes, 2nd edition, Springer.

Evaluation of Students: This is a seminar course, that is, the students will present the material, in approximately two lectures at a time. I will also assign some (very light) homework each week. Students will be graded on their lectures, homework assignments, and class participation.

The SFSU Algebra-Geometry-Combinatorics Seminar hosts many leading experts on polytopes. The seminar meets Fridays at 3:10.

"Philosophy is written in this grand book--I mean the universe--which stands continually open to our glaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it."
Galileo Galilei (Il Saggiatore, 1623)

more pearls of wisdom