Discrete and Polyhedral Geometry
Pak, MS 6240, pak@math.
Class Schedule: MWF 1:00-1:50, MS 6627.
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
- 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
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
HW1 (due April 30 in class)
HW2 (due May 14 in class)
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.
to return to Igor Pak Home Page.
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Last updated 8/14/2008