# Jerusalem Mathematics Colloquium

Thursday, 28th October 2004, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Professor Igor Pak

(MIT and Hebrew University)

"Convex polytopes, rigidity, and classical geometry"

** Abstract: **
Cauchy Theorem states that simplicial polytopes in three
dimensions are rigid, so in principle one should be able
to "construct" a polytope from its graph and edge lengths.
Actually doing this is more complicated even in the most
simple special cases...

In this talk we review several approaches to the problem.
We start with the Alexandrov "existence theorem", then
switch to a remarkable Kapovich-Millson "universality
theorem" on planar linkages, and then outline Sabitov's
polynomials for nonconvex polyhedra. The latter work
is related to Sabitov's proof of the "bellows conjecture"
that flexible polyhedrons must keep its volume constant
under continuous deformation. We conclude with joint
results of Fedorchuk and myself on the degrees of Sabitov
polynomials and sketch our proof of the Robbins conjecture
on the area of cyclic polygons.

The talk should be accessible to anyone who have seen
icosahedron, or at least heard about it...

Light refreshments will be served in the faculty lounge at 3:30.

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