Cornell Math - MATH 762, Spring 2005

MATH 762: Seminar in Geometry (Spring 2005)

Instructor: Robert Connelly

Meeting Time & Room

This is an introduction to the geometry of points and distances with applications to and from the theory of rigid and non-rigid structures. A basic role of geometry in science and mathematics is to determine when distance constraints on a configuration of points determine the configuration itself. This is connected to the theory of frameworks as used in engineering and well as distance geometry in mathematics.

Prerequisites: A good background in linear algebra (including matrices, determinants, symmetric matrices, eigen vectors, etc.) and some basics of calculus.

Topics:

  1. A classification of the congruences of Euclidean space.
  2. Infinitesimal and static rigidity of frameworks and tensegrities
  3. Infinitesimal rigidity implies rigidity
  4. Stresses and spider webs
  5. Applications to glasses, protein structure, and rigid membranes with holes
  6. Cauchy's Theorem abut the rigidity of convex polyhedra
  7. The stress-energy quadratic form/mathix
  8. Super stability and global rigidity
  9. Applications to the packing of congruent spherical balls and their stability
  10. The carpenter's rule problem about opening a piece-wise linear embedded arc.
  11. The Kneser-Poulsen problem about the areas of circles whose centers are contracted.