**Math 762 Home Page**

**Discrete Geometry: Rigid and Flexible Structures
- Spring 2001**

**Lecturer**:

**Lecture: **Tuesday, Thursday** **from 1:25 to 2:45 PM in Malott
230

**Course notes:** without pictures in pdf
format and in postscript
format.

**Texts: **We will mostly use notes handed out in class. The following
texts, which are on reserve in the Math library in Malott Hall, may be
helpful for related material.

- Linear Algebra, by Kenneth Hoffman and Ray Kunze, Prentice-Hall (1961)
- Matrix Analysis, by Roger A. Horn and Charles R. Johnson, Cambridge University Press (1985)
- Modern Graph Theory, by Béla Bollobas, Springer (1991)
- Convex Polytopes, by Branko Grünbaum, John Wiley (1967)
- Lectures on Polytopes, by Günter M. Ziegler, Springer (1998)
- Representations and Characters of Groups, by Gordon James and Martin Liebeck, Cambridge Math. Textbooks (1993)
- Combinatorial Rigidity, by Jack Graver, Brigitte Servatius and Herman Servatius, American Mathematical Society (1993)

**Problems:** We will have problems that will be due a week after
they are assigned. The first set of (two) problems was handed out on Thursday
1/25 and is due on Thursday 2/1.

Problem set 2: Due February 15. But if you need a hint or two, the hints will be given on February 12. If you do not need a hint, then you can hand in what you can do on February 12.

Problem set 5: Due March 1.

Problem set 6: Due March 8.

Problem set 7: Due March 15.

Problem set 8: Due March 29.

Problem set 9: Due April 5.

Problem set 10: Due April 12.

Problem set 11: Due April 19.

Problem set 12: Due April 26.

Problem set 13: Due May 3.

**Projects:** Instead of having a final exam, we can have the students
report on some work that is related to rigidity. Here are a few suggestions.

- Bracing grids: How do you put cross braces in a square grid to make it rigid? See: 81j:73066a Bolker, Ethan D.; Crapo, Henry Bracing rectangular frameworks. I. SIAM J. Appl. Math. 36 (1979), no. 3, 473--490. (Reviewer: Colin J. H. McDiarmid) 73K99 (05B35)
- Rigidity theory applied to glasses: See the book: Rigidity Theory and Applications, by M. F. Thorpe and P. M. Duxbury, Kluwer (1999).
- The Colin de Verdière number for a graph and the stress matrix: I have a recent preprint and the following may help: 2000h:05064 Lovász, L.; Schrijver, A. On the null space of a Colin de Verdière matrix. Symposium à la Mémoire de François Jaeger (Grenoble, 1998). Ann. Inst. Fourier (Grenoble) 49 (1999), no. 3, 1017--1026. (Reviewer: B. Zelinka) 05C10 (05C50)
- Polyhedral combinatorics: Rigidity theory can say some non-trivial things about the number of facets of a convex polytope. See 88b:52014 Kalai, Gil Rigidity and the lower bound theorem. I. Invent. Math. 88 (1987), no. 1, 125--151. (Reviewer: P. McMullen) 52A25 (57Q15)
- Higher-order rigidity: The natural definition is not as one might imagine. See 94m:52027 52C25 Connelly, Robert(1-CRNL); Servatius, Herman(1-MIT-AM) Higher-order rigidity---what is the proper definition?
- Rings of molecules in chemistry can be described with the help of the theory of rigid and flexible frameworks. See Chapter 4 in Distance Geometry and Molecular Conformation, by G. M. Crippen and T. F. Havel, Wiley (1988).
- Electrical networks and static rigidity: There is a close analogy between these two theories. For starters see Chapter 2 in 99h:05001 Bollobás, Béla Modern graph theory. Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998. xiv+394 pp. ISBN: 0-387-98488-7 (Reviewer: Jerrold W. Grossman) 05-01 (05-02 05Cxx).
- The Gale transform in convexity and the stress matrix: There is a relation between these two objects. I have some notes that explain it.
- The rigidity of stable packings and volume increasing motions: See 99c:52027 Bezdek, A.; Bezdek, K.; Connelly, R. Finite and uniform stability of sphere packings. Discrete Comput. Geom. 20 (1998), no. 1, 111--130. (Reviewer: S. Stein) 52C17 (52C25).
- The pebble game in "Rigidity Theory and Applications", edited by M. F. Thorpe and P. M. Duxbury, Kluwer (1999). See page 247.

Last updated: *April 19, 2001*