# Discrete and Polyhedral Geometry

**UCLA Course:** Math 285N, Spring 2021.

**Instructor:** Igor Pak

(see email instructions on the bottom of the page).

**CCLE Website:** is here.

**Class schedule:** MWF 2:00 - 2:50 pm, via Zoom.

Zoom meeting id: 998 7682 8512 (the full link will be posted on CCLE cite)

The password will also be sent by email from the course my.ucla site.

**Office Hours:** M 3:00-3:50, via Zoom (same link).

**Grading:** The grade will be based on attendance, class participation (20%),
and homeworks (80%) which will be posted below.

## 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

### Course reading:

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.

**Collaboration policy:**

For the home assignments, you can form discussion groups of up to 3 people each. In fact, I would like
to encourage you to do that. You can discuss problems but have to write your own separate solutions.
You should write the list of people in you group on top of each HA.

Click here
to return to Igor Pak Home Page.

To e-mail me, click
here and delete .zzz

Put *Math 285* in the Subject line.

*Last updated 3/21/2021*.