Convex polytopes

UCLA Course: Math 206A, Fall 2023.

Instructor: Igor Pak
(see email instructions on the bottom of the page).

CCLE Website: is here.

Class schedule: MWF 11:00 - 11:50 pm, MS 5203.

Office Hours: M 2:00-2:50.

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

Content:

• 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
• Dehn-Sommerville equations, Kalai's "simple way to tell a simple polytope", Balinski theorem
• 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
• Alexandrov-Fenchel inequality, equality conditions
• Complexity aspects

Course reading:

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.

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 206A in the Subject line.

Last updated 10/2/2023.