## MATH 7620: *Seminar in Geometry * (Spring 2009)

**Tuesday, Thursday 2:55-4:10, Malott 205**

**Instructor:** Robert Connelly
I would like to cover the fundamentals of the theory of discrete rigid
structures leading to subjects of recent interest in generic global rigidity.
I would like to cover a subset of the following, along with open problems,
the amount depending how time permits:

**Links and Texts:**

- Maria Belk's handwritten notes. Rigidity
Notes Part I (8.73 MB). Rigidity
Notes Part II. (13.37 MB)

- Igor Pak's book. Lectures on Discrete
and Polyhedral Geometry. (3 MB)
- My survey "Rigidity" (10.27 MB)
in Handbook of convex geometry, Vol. A, B, 223--271, North-Holland, Amsterdam,
1993.
- My article "Rigidity and Energy"
(1.03 MB), Invent. Math. 66 (1982), no. 1, 11--33.
- My article with Walter Whiteley, "Second-order
rigidity and prestress stability for tensegrity frameworks" (4.43 MB),
SIAM J. Discrete Math. 9 (1996), no. 3, 453--491.

Problem Sets:
- Due February 3, 2009.
- Due February 17, 2009.
- Due February 24, 2009.
- Due March 3, 2009.
- Due 2009.

**Basics**

- Rigid and flexible frameworks including tensegrities: definitions
and motivation.
- Global rigidity
- Infinitesimal and static rigidity.
- Cauchy's Theorem and Dehn's Theorem about rigid polyhedra in 3-space.
- Laman's Theorem about generic rigidity in the plane.
- The pebble game: an algorithm for generic rigidity in the plane.

### The Stress Matrix

- The motivation and definition of the stress matrix.
- Applications of the stress matrix to the global rigidity of tensegrities.
- Applications of the stress matrix to generic global rigidity.

### Symmetry

- A quick intro to representation theory.
- Applications of representation theory to the calculation of the stability
and global rigidity of symmetric tensegrity structures.
- An introduction to my catalog (with R. Terrell and A. Back) of the
combinatorial types of symmetric tensegrities.

### Carpenter's Rule Theory

- An introduction to the problem of opening a polygon in the plane.
This is an application of the basic theory.
- An introduction to pseudotriangulations following I. Streinu's work.

### Kneser-Poulsen Theory

- An introduction to the problem of calculating the change in the area/volume
of unions, intersections, etc. of finite sets of disks.
- A description of Csikos's formula for the change in such a union etc.
of disks.
- Possible applications of Csikos's formula to extremal problems involving
finite collections of overlaping disks.

*Last modified: March 6, 2008*