Discrete and Polyhedral Geometry

Instructor: Igor Pak, MS 6240, pak@math.

Class Schedule: MWF 1:00-1:50, MS 6627.

Brief outline

We will give an introduction to the subject, covering a large number of classical and a few recent results. The emphasis will be on the main ideas and techniques rather than proving the most recent results in the field. The idea is to to give a guided tour over (large part of) the field to prepare for more advanced results in the future.

The prerequisites for the course are relatively small: some advanced undergraduate linear algebra and discrete mathematics (enumerative combinatorics and basic graph theory). On several occasions I will use basic tools and ideas from algebra, but in each case I will fully explain what is needed so the course will be largely self-contained.

The lectures will follow selected chapters from my forthcoming book.

Content:

Helly theorem. Various extensions and generalizations.
Barany theorem. The planar case via fair division.
Dehn-Sommerville equations, Kalai's "simple way to tell a simple polytope", Balinski theorem
Triangulations of polygons and polyhedra, local move connectivity
Scissor congruence in the plane, Hilbert third problem (Bricard's version and Kagan's presentation of the Dehn invariant)
Polytope algebra and Sydler's theorems
Cauchy and Dehn theorems, examples of flexible polyhedra
Proof of the bellows conjecture (after Connelly, Sabitov, and Walz)
Alexandrov theorems on polytopes with vertices on rays and given curvature, Pogorelov's proof
The Brunn-Minkowski inequality and the Minkowski theorem on polytopes
Nonoverlapping unfoldings of convex polytopes

Grading:

If you are taking this course for credit, there will be about 4 homeworks with several challenging problems. The homeworks will be posted on this page.

HW1 (due April 30 in class)

HW2 (due May 14 in class)

Course Textbooks:

My book should suffice. For further reading see:

J. Matousek, Lectures on Discrete Geometry, Graduate Texts in Mathematics 202, Springer, 2002.
G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, 1995.
P.M. Gruber, Convex and discrete geometry, Springer, Berlin, 2007.
A. Barvinok, Course in Convexity, Graduate Studies in Mathematics 54, AMS, 2002.
B. Grunbaum, Convex Polytopes, Graduate Text in Mathematics 221, Springer, 2003.
J. Pach and P.K. Agarwal, Combinatorial geometry, John Wiley, New York, 1995.

I believe all these books are available in the math library, from Amazon.com and other retailers.

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Last updated 8/14/2008