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.