# Discrete and Polyhedral Geometry

Course: MATH 8680, Topics in Combinatorics

Instructor: Igor Pak, VinH 258, pak@math.

Class Schedule: MWF 2:30, VinH 211.

## Brief outline

We will discuss a variety of combinatorial and geometric questions on convex polytopes, along with a number of application. The content of the course is outlined below. 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
• Variational approach (billiard trajectories, closed geodesics)
• Steinitz theorem, outline of two proofs
• Perles's example of an "irrational polytope", and the Lawrence construction
• 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
• Gauss-Bonnet theorem for convex polyhedra
• Cauchy Theorem (proofs by Legendre-Cauchy, Dehn, Pogorelov-Volkov)
• 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
• Alexandrov's uniqueness and existence theorems characterizing polytopes by their unfoldings
• The Brunn-Minkowski inequality and the Minkowski theorem on polytopes
• Nonoverlapping unfoldings of convex polytopes
P.S. This is a little too ambitious. Due to the time constraints I might have to cut a few results at the end.

### 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.