# 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*