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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:
- A classification of the congruences of Euclidean space.
- Infinitesimal and static rigidity of frameworks and tensegrities
- Infinitesimal rigidity implies rigidity
- Stresses and spider webs
- Applications to glasses, protein structure, and rigid membranes with
holes
- Cauchy's Theorem abut the rigidity of convex polyhedra
- The stress-energy quadratic form/mathix
- Super stability and global rigidity
- Applications to the packing of congruent spherical balls and their
stability
- The carpenter's rule problem about opening a piece-wise linear embedded
arc.
- The Kneser-Poulsen problem about the areas of circles whose centers
are contracted.
Last modified:
October 4, 2004
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