MATH 7620: Seminar in Geometry (Spring 2009)

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:

1. Maria Belk's handwritten notes. Rigidity Notes Part I (8.73 MB). Rigidity Notes Part II. (13.37 MB)
   
2. Igor Pak's book. Lectures on Discrete and Polyhedral Geometry. (3 MB)
3. My survey "Rigidity" (10.27 MB) in Handbook of convex geometry, Vol. A, B, 223--271, North-Holland, Amsterdam, 1993.
4. My article "Rigidity and Energy" (1.03 MB),  Invent. Math. 66 (1982), no. 1, 11--33. 
5. 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.

1. Due February 3, 2009.
2. Due February 17, 2009.
3. Due February 24, 2009.
4. Due March 3, 2009.
5. Due 2009.
   

Basics

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

The Stress Matrix

1. The motivation and definition of the stress matrix.
2. Applications of the stress matrix to the global rigidity of tensegrities.
3. Applications of the stress matrix to generic global rigidity.

Symmetry

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

Carpenter's Rule Theory

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

Kneser-Poulsen Theory

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