Researchers

Mathematics / CONNELLY, ROBERT

Connelly

Position

Professor and chair in the Department of Mathematics at Cornell University.

Research interests

Discrete geometry, computational geometry and the rigidity of discrete structures.

Discrete geometry, with emphasis on the geometry of rigid and flexible structures, is his main area of interest. A tensegrity is a structure composed of sticks held in mid-air with strings which, nevertheless, holds its shape. This can be modeled very nicely as a configuration of points with upper and lower bounds on the distances between certain pairs of points. This in turn leads to interesting problems in, and applications to, distance geometry and the theory of packings and coverings of spheres as well as applications to robotics, protein folding, motion planning and percolation problems in physics and probability.

Another subject of interest is the theory of flexible surfaces. There are triangulated surfaces that flex, keeping their edges at a fixed length, and it has recently been shown that such surfaces maintain a fixed volume while they flex. There is no perfect mathematical bellows. This is also related to a polynomial that relates the volume of the surface to the lengths of its edges. This is at the intersection of discrete geometry, algebraic geometry and topology.

Prizes and awards

Publications

Robert Connelly with Maria Sloughter, Realizability of Graphs, 2004.

Robert Connelly with Joseph Zaks, The Beckman-Quarles theorem for rational $d$-spaces, $d$ even and $d\geq 6$, Discrete geometry, 193--199, Monogr. Textbooks Pure Appl. Math., 253, Dekker, New York, 2003.

Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato and P. M. Chaikin, ''Improving the Density of Jammed Disordered Packings using Ellipsoids'', Science, 303:990-993, 2004.

The Kneser-Poulsen conjecture for spherical polytopes. Discrete Comput. Geom. 32 (2004), no. 1, 101--106.

Pushing disks apart---the Kneser-Poulsen conjecture in the plane. with K. Bezdek J. Reine Angew. Math. 553 (2002), 221--236.

Generic Global Rigidity, (to appear in Discrete Comput. Geom.).

Comments on Generalized Heron Polynomials and Robbins' Conjectures.

''A Linear Programming Algorithm to Test for Jamming in Hard-Sphere Packings'', by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, J. Comp. Phys, 197 (1):139-166, June 2004.

''Straightening Polygonal Arcs and Convexifying Polygonal Cycles" (joint with Erik Demaine and Günter Rote) in Discrete and Computational Geometry, Vol. 30, No. 2, (Sept. 2003), 205-239).

''Two-distance preserving functions from Euclidean space" (joint with Károly Bezdek) Periodica Mathematica Hungarica Vol. 39 (1-3), (1999), pp. 185-200.

'' The Bellows Conjecture'', joint with I. Sabitov and A. Walz in Contributions to Algebra and Geometry , volume 38 (1997), No.1, 1-10.

''Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks", (joint with Walter Whiteley, SIAM J. Discrete Math, Vol. 9, No. 3, pp. 453-491, August 1996.