## Welcome!

Planning has begun for our Conference. The purpose is to bring together mathematicians and scientists who are already working in the area of rigidity and stability of structures. Information will be posted here as it becomes available.

Unfortunately we will not have travel funds for our participants, but expect to have an interesting collection of speakers and topics. More to come soon.

We will have some, pastries, sandwiches, coffee, etc. for breakfast and lunch in the fifth floor lounge in Malott Hall.

You need a parking permit to park on campus. See Bob Connelly for information about how to buy such a permit. See here for info.

The talks will be in Malott 251 on the ground floor in the Annex.

### Speakers, Title, Abstract

- Bob Connelly, Cornell, Conference Overview and Packing Problems (This will attempt to connect packing problems and their rigidity to the combinatorics of triangulated packings. One application of this point of view is a slolution to the isostatic conjecture in granular materials.)
- Alexey Garber, Texas, On rigidity of certain manifolds constructed from squares (Let $M$ be a two-dimensional manifold with associated cell-decompostion such that all two-dimensional cells are quadrilaterals. Assume that for such a cell complex there exists a map $f$ from $M$ to $\mathbb{R}^3$ such that image of each quadrilateral is a unit square. For example, the surface of unit three-dimensional cube can be constructed in this way if we start from $M$ being $\mathbb{S}^2$. In 1990s Dolbilin, Shtan'ko and Stogrin proved that under certain local conditions on $f$ the image $f(M)$ is rigid provided $M$ is an orientable surface of small genus. In this talk we will give a simple proof of these results and discuss similar questions in higher dimensions. Based on a joint work with Alexey Glazyrin.)

- Steven Gortler, Harvard, Reconstruction from unlabled path lenghts (Our community often focuses on questions about the uniqueness (global rigidity) and determination (localization) of a point configuration, given a set of edge lengths, where each length is labeled with the combinatorial edge that it corresponds to. Less studied is the situation where we are only given the unlabeled length data. This problem appears in a variety of practical applications. I will talk about some of these unlabeled rigidity questions in general. I will in particular talk about the situation where the data includes not just unlabeled edge lengths, but also unlabeled path lenghts (a path begin some sequence of edges).
- Simon Guest, Cambridge, 'Bending' Tesnegrities (Tensegrities are traditionally made up of elements that carry loads axially along their length. But there are examples of structures where bodies which carry 'bending' loads are held apart from one another by cables in tension. These examples share many characteristics of traditional tensegrities, such as requiring a state of self-stress to give rigidity. The talk will show some examples of these structures, and start to consider how they might be analysed.)
- Oleg Musin, Texas, Rigid congruent spherie packings (We consider contact graphs of congruent sphere packings in Euclidean and spherical spaces. Recently, we enumerate up all locally rigid circle packings on the unit sphere with number of circles N<12. This problem is equivalent to the enumeration of irreducible contact graphs. In this talk we show that by using the list of irreducible graphs can be solved various problems of extreme packings such as the Tammes and the maximal contacts problems.)

- Tony Nixon, Lancaster, Generic global rigidity on the cylinder (A fundamental theorem in rigidity theory states that a generic framework in 2-dimensions is globally rigid if and only if the underlying graph is a complete graph on at most four vertices or is both redundantly rigid and 3-connected. I will show an analogue for frameworks in 3-dimensions whose vertices are constrained to lie on the cylinder. The proof of this makes use of: a new recursive construction of circuits in the simple (2,2)-sparse matroid; a characterisation of rigidity for generic frameworks on the cylinder with a designated vertex allowed to move off the cylinder; and a characterisation of both rigidity and global rigidity for generic frameworks on the cylinder with the added constraint that the projection of a motion onto the axis of the cylinder should be a dilation.)
- Mahdi Sadjadi, Arizona, Tunneling Modes in Glasses and Jammed Systems (We employ a theorem that says that isostatic systems must have an even number of realizations [1]. This means that while a realization can be locally rigid, they cannot be globally rigid. This provides a likely explanation for tunneling modes in glasses [2] that have been known to exist experimentally for a long while but have never been satisfactorily explained from a structural viewpoint. When a constraint is removed from an isostatic system, a single degree of freedom is introduced that results in a continuous circuit which introduces at least one new realization, where the original constraint is restored [3]. While cutting a single edge is one example of this in large systems, this leads to very few tunneling modes, and more complex schemes are discussed. Computer algorithms are optimized for time by taking fairly large steps, as determined by the eigenvalue from the dynamical matrix, and then doing a very accurate relaxation once the system is in the catchment area of the new realization. We discuss results for this procedure for pinned, free standing and periodic systems. Results are presented for two-dimensional systems of both corner sharing triangles and jammed systems of discs [4]. While tunneling modes are expected in glassy networks, they are a surprise for jammed systems of discs and the implications are discussed. It is also important to find the height of the barrier in going from one state to another [5].)
- Bernd Schulze, University of Lancaster, Global rigidity of periodic frameworks (A d-dimensional (bar-joint) framework is `globally rigid' if it is the only framework in d-space with the same graph and edge lengths, up to congruent motions. In 1992 Hendrickson established necessary conditions for a generic framework to be globally rigid in d-space. In 2005 Connelly, Jackson and Jordan showed that these conditions are also sufficient for a generic framework in the plane to be globally rigid. In this talk we present analogues of Hendrickson's necessary conditions for generic periodic frameworks with a fixed lattice representation (or, equivalently, generic finite frameworks on a flat torus of fixed size and shape) to be globally rigid in d-space. For d = 2, we then show that these conditions are also sufficient for generic global rigidity. This provides a combinatorial characterisation of the periodic graphs (graphs on the fixed torus) whose generic realisations are globaly rigid in the plane.)

- Meera Sitharam, University of Florida, Materials configuration spaces (TBA)
- Evan Solomonides and Maria Yampolskaya, Cornell, Disk Packing Simulation (The program used in our research (Professor Connelly's recent projects) will be demonstrated and explained. Talk to be given with Maria Yampolskaya.)

- Ken Stephenson, Kentucky, Configurations of circles: beyond packings (We survey the study of configurations of circles inspired by Bill Thurston's circle packings. However, ours will be inversive distance packings, which involve configurations with prescribed overlaps, tangencies, and/or separations between circles. Their study provides an intriguing mash up of theory, computation, experimentation, and application. We discuss existence, formal rigidity, and relaxed notions of rigidity associated with discrete conformality and convergence behavior. Examples will be drawn from concrete phenomena in nature and emerging applications.)

- Louis Theran, St. Andrews, TBA
- Brigitte Servatius, WPI, Combinatorial Maps, Delta Matroids and Rigidity

### Organizers

- Bob Connelly (chair)
- Steven Gortler, Harvard
- Mike Thorpe, University of Arizona at Tempe
- Miranda Holmes-Cerfon, NYU
- Meera Sitharam, University of Florida
- Simon Guest, Cambridge University, England

### Conference Coordinator

- Heather Peterson