Bill Jackson - Rigidity of graphs and frameworks скачать в хорошем качестве

Bill Jackson - Rigidity of graphs and frameworks

This talk was part of the Workshop “Rigidity and Flexibility of Geometric Structures” held September 24 -28, 2018 at the ESI. The first reference to the rigidity of frameworks in the mathematical literature occurs in a problem posed by Euler in 1776. Consider a polyhedron P in 3-space. We view P as a ‘panel-and-hinge framework’ in which the faces are 2-dimensional panels and the edges are 1-dimensional hinges. The panels are free to move continuously in 3-space, subject to the constraints that the shapes of the panels and the adjacencies between them are preserved, and that the relative motion between pairs of adjacent panels is a rotation about their common hinge. The polyhedron P is rigid if every such motion results in a polyhedron which is congruent to P. Euler’s conjecture was that every polyhedron is rigid. The conjecture was verified for the case when P is convex by Cauchy in 1813. Gluck showed in 1975 that it is true when P is ‘generic’ i.e. there are no algebraic dependencies between the coordinates of the vertices of P. Connelly finally disproved the conjecture in 1982 by constructing a polyhedron which is not rigid. I will describe results and open problems concerning the rigidity of various other types of frameworks. I will be mostly concerned with the generic case for which the problem of characterizing rigidity usually reduces to pure graph theory. 1. Lead 00:00:00 2. Bar-joint frameworks 00:00:22 3. Infinitesimal motions. Generic rigidity 00:10:56 4. Maxwell's condition and Laman's Theorem 00:22:59 5. The Dress conjecture 00:37:28 6. General global rigidity. Stress Matrix 00:44:52 7. Different kinds of frameworks 00:51:28 8. Questions from the audience 01:00:50