**Instructor: ** Igor
Pak, 2-390, pak@math.

**Class Schedule: ** Tu,Th 11-12:30, MIT Room 4-265
(see here to find it).

- Kissing number of a sphere is < 14 (in fact it's 12 but we won't prove that)
- Helly Theorem. Various extensions and generalizations.
- Dehn-Sommerville equations, Kalai's "simple way to tell a simple polytope"
- Perles's example of an "irrational polytope", Mnev's Universality Theorem via Lawrence construction
- Linkages, theorems of Kempe and Kapovich-Millson (without proof)
- Cauchy Theorem (geometric and algebraic proofs)
- Steinitz Theorem, rationality of 3-dim polytopes
- Sabitov polynomials for polyhedra homeomorphic to a sphere (after Connelly, Sabitov, and Walz)
- dimension of moduli spaces of polytopes, bipyramids, solution of the Robbins conjecture
- Alexandrov "existence" theorem, Volkov's approach
- nonoverlapping unfoldings, what happens in higher dimensions
- Brunn-Minkowski inequality, applications to order polytopes
- Minkowski Theorem on polytopes
- Koebe Theorem on circle packing, application to plane separator theorem
- counting integer points on polytopes, Barvinok's algorithm

*P.S.* I know, I know - it's a little too ambitious - the final
cut will be made later...

**Course Textbooks:**

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.

Both are available at
Quantum Books

**More Textbooks:**

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.

- For background on kissing numbers see
and Conway and Sloane,
*Sphere Packings, Lattices and Groups*. For an interesting and (relatively) simple proof of*k*_{3}= 12 see this recent paper by Oleg Musin. - For Helly Theorem, various extensions and generalizations see
classical survey article Danzer, Grunbaum, and Klee,
*Helly Theorem and its relatives*(1963). In our presentation we followed chapters 8 and 9 of Matousek.For classical introduction to Borsuk theorem (case

*d*= 2) and Borsuk conjecture see Hadwiger and Debruner,*Kombinatorische Geometrie In Der Ebene*. For disproof of the conjecture see Kahn and Kalai original article. Compare with the version in Aigner and Ziegler,*Proof from the Book*. - For Dehn-Sommerville equations and Kalai's simple way to
tell a simple polytope see Ziegler, sections 8.3 and 3.4. Our
presentation of DS-equation followed Bronsted,
*An introduction to convex polytopes*. - For Balinski theorem see Ziegler, sections 3.5.
Our presentation of the
*Y*&Delta proof of Steinitz theorem followed Ziegler's book, sections 4.1-4.3. For a classical approach and various connections see Grunbaum, chapter 13. For Tutte's equilibrium approach see chapters 12, 13 in this book by Richter-Gebert (it was published in Springer's Lecture Notes).

For Koebe-Andreev-Thurston theorem and its applications see Pach-Agarwal book. The variational approach we presented followed section 1 in recent Ziegler's lectures notes

- For Mnev's Universality theorem, various extensions, and references
see above mentioned Richter-Gebert's
book.
An introduction and the Lawrence construction is given in
Ziegler, section 6.6. Note also a good (and very short) discussion
in Matousek, end of section 5.3.
The polytope of Micha Perles is described in Grunbaum (section 5.5) and in Ziegler (section 6.5).

- For a background on linkages see classical books Hilbert and Cohn-Vossen,
*Geometry and the Imagination*, chapter 5; Courant and Robbins*What is Mathematics?*, section 3.5.For modern approach and Kapovich and Millson's theorem see their original article; see also King's followup.

Click here to return to Igor Pak Home Page.

To e-mail me click
here and delete .zzz

Put *18.318* in the Subject line.

*Last updated 1/27/2005*