Math 762 Home Page

Discrete Geometry: Rigid and Flexible Structures - Spring 2001


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

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

Course Description:

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.

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.

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.

  1. 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)
  2. Rigidity theory applied to glasses: See the book: Rigidity Theory and Applications, by M. F. Thorpe and P. M. Duxbury, Kluwer (1999).
  3. 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)
  4. 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)
  5. 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?
  6. 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).
  7. 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).
  8. The Gale transform in convexity and the stress matrix: There is a relation between these two objects. I have some notes that explain it.
  9. 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).
  10. 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