# Rigidity and polyhedral combinatorics

December 3 to December 7, 2007

at the

American Institute of Mathematics, Palo Alto, California

organized by

Robert Connelly, Ezra Miller, and Igor Pak

This workshop, sponsored by AIM and the NSF, will be devoted to polyhedral objects in Euclidean spaces. Specifically, which metric or combinatorial properties must change or remain constant under certain classes of deformations, such as bending, folding, stretching, or flexing? When it is impossible to carry out one or another of these operations in some prescribed manner, an object is said to possess some sort of "rigidity". This can occur, for example, when too many metric or combinatorial properties are forced to remain constant under a given operation. The past several years have seen a number of significant advances, such as a proof of the bellows conjecture, constructions of unfoldings for polytopes of all dimensions, and an algorithm for Alexandrov's theorem on 3-polytopes. But with those advances has come increasing recognition that many basic questions remain largely open, particularly in higher dimensions.

1. Geometry and algebra of flexible objects.
In 1977, R. Connelly showed that there exists a surface in Euclidean three-space composed of rigid triangles, hinged along their edges, that is flexible. In 1995 I. Sabitov showed the very remarkable "bellows property" for such flexible surfaces: the volume they bound is constant during the motion; thus there is no perfect mathematical bellows. But even more remarkable is that there is a monic polynomial, whose coefficients are themselves polynomials in the edge lengths, which is satisfied by the volume. This is a new fundamental property of triangulated surfaces in Euclidean three-space; it is an algebraic identity, but it does not seem to be part of what is known in algebraic geometry. Another example is the area of a cyclic polygon in the plane, which is integral over the ring generated by the edge lengths (a favorite problem studied by David Robbins just before his death, and solved just after his death by Fedorchuk-Pak and others). These suggest problems such as the following, some of which could benefit from key insights using algebraic geometry of configuration spaces -- perhaps the kind of algebraic geometry considered by Develin, Martin, and Reiner in their work on rigidity.
1. Is there a higher-dimensional analogue of the Sabitov polynomial? Only partial results are known here.
2. Are there other geometric quantities that are integral over rings generated by other geometric quantities? For example, is the n-dimensional volume of an n-simplex integral over the ring generated by the k-dimensional volumes of its k-faces for k=2, n=5?
3. Are there non-trivial examples of flexible triangulated surfaces for dimensions greater than 4?
4. With the example of flexible surfaces, which are known, not only is the volume constant, but the surface seems to be the sum of rigid surfaces and flexible surfaces of zero volume, where the surfaces are regarded as singular simplicial cycles in the sense of algebraic topology. Is this always true?
5. For flexible surfaces, is the Dehn invariant constant during the motion? If so, this would imply that the regions bounded by the surfaces are equivalent by dissection, as in Hilbert's third problem.

2. Folding and bending.
There is a tradition, centering around the school of A. D. Alexandrov in Russia, of interest in the rigidity of convex polyhedral surfaces in three-space. The issues are related to the classical theorem of Cauchy in 1813 that says there is a unique convex realization of any convex polyhedron in three-space once the the faces and their adjacencies are determined. Alexandrov himself showed a basic existence result: any intrinsically convex polyhedral surface can be realized as an extrinsically convex polyhedron or doubly covered polygon in three-space. Recently, Bobenko and Izmestiev have shown how to implement this result by an effective new proof of Alexandrov's theorem. Their methods produce an explicit algorithm to obtain any three-dimensional polytope by appropriately gluing any given foldout ("development") of its boundary in the plane. Many intriguing questions related to this theory remain.
1. Given a convex polyhedral surface, can one cut along some of the edges and unfold it into a polyhedral disk in the plane? The analogous question makes sense in higher dimensions, but even for polyhedra in three-space it is open. Miller and Pak have related results in all dimensions, but the cuts are not restricted to lie along ridges (edges, in the case of surfaces).
2. Can one unfold polyhedral surfaces in three-space expansively (continuous blooming)? Does the source unfolding bloom in higher dimensions?
3. J-M Schlenker has a conjecture that extends Cauchy's result to the case when the vertices only are in convex position, and the interior can be triangulated without adding more vertices. It would be interesting to look at some cases to check the conjecture.
4. What is the maximum volume bounded by a surface submetric to a given polyhedral surface? Polyhedral surfaces never bound the maximum volume, by a theorem of I. Pak, but can approximate it; what does the limiting surface look like? For example, is it smooth except at finitely many points?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.