Abstracts

Plenary Speakers

Miranda Holmes-Cerfon (Courant Institute, NYU)

Rigidity theory in statistical mechanics (Holmes slides Bond-node structures 2017)

One long-standing puzzle in statistical mechanics is to explain how materials crystallize - in three dimensions, small systems seem to prefer different arrangements from large ones, a phenomenon known as geometric frustration. I will look at geometric frustration in colloids, particles which form the building blocks of many common materials, and show how ideas from rigidity theory bring insight into frustration which is difficult to obtain using traditional methods from statistical mechanics. In particular, I will show how non-generic or singular frameworks, those where the number of infinitesimal degrees of freedom does not equal the number of finite degrees of freedom, play a critical role in explaining certain observations. In addition, I will describe some ongoing work (and challenges) developing algorithms to efficiently explore configuration spaces of singular frameworks.

Tibor Jordan (Eotvos Lorand University)

Globally rigid braced triangulations (Jordan slides Bond-node structures 2017)

Cauchy proved that if the vertex-edge graphs of two convex polyhedra are isomorphic and corresponding faces are congruent then the two polyhedra are the same. It follows that a convex polyhedron with triangular faces, as a bar-and-joint framework, is rigid. It is also well-known that the graph of such a polyhedron - a maximal planar graph - is rigid in three-space, provided it is realised in a sufficiently general position.

Global rigidity is stronger than rigidity: a framework is said to be globally rigid if the bar lengths determine all pairwise distances between the joints. The graphs of triangulated polyhedra are not globally rigid. We shall consider - the graphs of - braced triangulated polyhedra and characterise the bracing patterns which make them globally rigid in three-space.

Joint work with Shin-ichi Tanigawa.

Meera Sitharam (University of Florida)

Materials wonderland through geometric constraints looking glass

To take advantage of the reality of 3D printing at the microscale, traditional geometric modelling needs to seamlessly incorporate material structural properties. Happily, as this talk will argue, much of the theory of geometric constraints - already heavily utilized in macroscale modelling - is well-suited to the microscale.

The talk will tour 3 scenarios: 2D layered materials, sphere-based self-assembled materials and supramolecular complexes. The goal of the talk is to illustrate how theorems, core algorithms and software implementations based on geometric constraint representations, combinatorial rigidity and configuration spaces are crucial for efficient modelling, analysis, mechanistic understanding and design of various material properties and phenomena: from distribution of density/stress/strain, fracture and cryptomorphic physical phenomena, to configurational entropy, kinetics and allostery.

Vincenzo Vitelli (Leiden University)

Active metamaterials with rotational constraints

Active metamaterials are built out of interacting components individually powered by motors. In this talk, we focus on chiral fluids that display active rotational motion or circulation. First, we show how to generate topological sound in fluids of self-propelled particles spontaneously circulating in annular-channel lattices. Next, we discuss an exotic transport coefficient, called odd viscosity, which controls the hydrodynamics of compressible fluids of active rotors.

Further Talks

We will also have 30-minute talks from:

Ciprian Borcea (Rider University)

Auxetic deformations and elliptic curves

In materials science and engineering, auxetic behaviour refers to deformations of flexible structures were stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behaviour for flexible periodic bar-and-joint frameworks.

Currently, the only known algorithmic solution is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper, we present a new, simpler algorithmic approach which is applicable to a natural family of three-dimensional periodic bar-and-joint frameworks with three degrees of freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve.

Bryan Chen (Massachusetts Amherst)

The geometry of flat origami triangulations (Chen slides Bond-node structures 2017)

Rigid origami structures consist of a network of folds and vertices joining unbendable plates. When all the faces are triangles, they can be modelled by a bond-node structure. While the generic configuration space of such a structure can be arbitrarily complicated, one can often gain insight into the allowed deformations by linearizing the geometric constraints. The singularity at flat configurations of origami limits the effectiveness of this approach. We study the infinitesimal second-order rigidity of flat origami triangulations, present numerical results and conjectures when their fold patterns are derived from Poisson-Delaunay triangulations and construct a family of origami where the singularity can be understood explicitly.

Joint work with Chris Santangelo.

Sean Dewar (Lancaster)

The rigidity of infinite frameworks in Euclidean and polyhedral normed spaces (Dewar slides Bond-node structures 2017)

In their paper "The rigidity of graphs" Asimow and Roth defined different definitions for infinitesimal and finite rigidity for finite bar-joint frameworks and ultimately showed that they are equivalent almost everywhere. However, the mathematics used to show this breaks down when infinite frameworks are considered. I will discuss how with the choice of a natural topology on the set of placements of an infinite framework we can obtain a similar equivalence of definitions for Euclidean spaces and a stronger equivalence for frameworks in polyhedral normed spaces (expanding the results of Derek Kitson in his paper "Finite and infinitesimal rigidity with polyhedral norms").

Georg Grasegger (RICAM/ JKU Linz)

Counting realizations of Laman graphs

We present a recently developed algorithm for computing the number of Euclidean realizations of minimally rigid graphs. These realizations can be considered as solutions of a system of algebraic equations with parameters for the lengths of the edges. The number of complex solutions does not depend on these parameters if the latter is chosen generically. The algorithm we present is based on a combinatorial recursion formula; its proof relies on algebraic and tropical geometry. Using the algorithm we computed the number of realizations for all Laman graphs up to 12 vertices.

Joint work with J. Capco, M. Gallet, C. Koutschan, N. Lubbes and J. Schicho.

Hakan Guler (Queen Mary)

2-dimensional rigidity with three coincident points (Guler slides Bond-node structures 2017)

Let G=(V,E) be a graph and u, v be distinct vertices of G. Let (G,p) be a framework such that p(u)=p(v) and this is the only algebraic dependency of p. Fekete, Jordan and Kaszanitzky characterised rigidity of such frameworks in 2-dimensions. In this talk, we give a characterisation for the frameworks in 2-dimensions when there are three distinct vertices u, v, w with p(u)=p(v)=p(w) and these are the only algebraic dependencies of p.

Joint work with Bill Jackson.

Bill Jackson (Queen Mary)

The rigidity of frameworks on a grid

Given a graph G=(V,E) and two partitions X and Y of V with the property that X_i and Y_j intersect in at most one vertex for all X_i in X and Y_j in Y, we characterise when G has an infinitesimally rigid realisation in the plane in which the vertices in each X_i in X lie on the same vertical line and the vertices in each Y_j in Y lie on the same horizontal line.

Joint work with John Owen and Steven Power.

Csaba Kiraly (Eotvos Lorand)

Sufficient conditions for the global rigidity of periodic graphs (Kiraly slides Bond-node structures 2017)

Tanigawa (2015) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result for periodic frameworks on a fixed lattice. A periodic framework is vertex-redundantly rigid if the deletion of a single vertex in each orbit of the lattice results in a periodically rigid graph. The proof is similar to the one of Tanigawa, however, there are some difficulties as follows. First, it is not known whether the periodic global rigidity is a generic property, however, by a slight modification of a recent result of Kaszanitzky, Schulze and Tanigawa (2016), this issue can be handled. The second problem is that, while the rigidity of graphs in R^d on at most d vertices obviously imply their global rigidity this is not obvious for periodic frameworks. We prove a similar result for periodic frameworks by extending a result of Bezdek and Connelly (2002) on the existence of a continuous movement between two equivalent d-dimensional realisations of a single graph in R^2d for periodic frameworks.

As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimensions by using a proof similar to the one of Tanigawa (2016). This extends the result of Connelly, Jordan and Whiteley (2013) from finite to infinite periodic frameworks.

Joint work with Viktoria E. Kaszanitzky and Bernd Schulze.

Derek Kitson (Lancaster)

Infinitesimal rigidity for unitarily invariant matrix norms (Kitson slides Bond-node structures 2017)

In this talk, we will consider infinitesimal rigidity theory for spaces of matrices endowed with a unitarily invariant matrix norm. Our primary goal is to obtain necessary counting conditions for graphs which admit an infinitesimally rigid placement in a given admissible matrix space. We will provide analogues of the Maxwell counting criteria for Euclidean bar-joint frameworks and show that minimally rigid graphs belong to the matroidal class of (k,l)-sparse graphs for suitable values of k and l.

Joint work with Rupert Levene.

Steve Power (Lancaster)

Double distance frameworks (Power slides Bond-node structures 2017)

We give characterisations of minimal generic rigidity for various frameworks which have two types of bonds/bars/constraints between their joints. Such Laman-type results require an understanding of the combinatorics of bicoloured graphs with mixed sparsity properties, together with an understanding of rigidity preservation under coloured construction moves, such as coloured Henneberg moves. Some potential application areas (NMR and sensor networks) are also indicated.

Joint work with Tony Nixon.

Hattie Serocold (Lancaster)

Coordinated isostatic frameworks (Serocold slides Bond-node structures 2017)

We define a coordinated framework to be a standard framework (G,p) with a colouring of the edges. Bars of the framework with the same colour are required to change length in a coordinated motion, all at the same rate, though some bars are permitted to remain uncoloured. We shall look at an inductive construction of isostatic coordinated frameworks, and characterise isostatic frameworks in 2-dimensions with 1 set of coordinated bars.

Brigitte Servatius (WPI)

Hilda Pollaczek-Geiringer's rigidity papers (Servatius slides Bond-node structures 2017)

Jan Peter Schafermeyer recently made the rigidity community aware of the work of Hilda Pollaczek-Geiringer, who published some papers on 2- and 3-dimensional trusses in the 1920s and '30s. We will examine her methods and compare them with the 1970 approach of Laman.

Ileana Streinu (Smith College)

From frameworks to molecules: the KINARI experience

KINARI (KINematics And RIgidity) is web-based software for rigidity and flexibility analysis of biomolecules and crystals, developed in the research group of the speaker at the University of Massachusetts Amherst and Smith College. The project started about 10 years ago and whose first release was made public in 2011, has helped our team gain useful insights into the intricacies of modelling molecules as mechanical frameworks. The lessons learned in turning mathematical ideas and algorithms into actual software, while working with real (incomplete, imprecise, noisy, ambiguous) molecular and crystal data have led to new mathematical and computational questions, which will be summarized in this talk. New features from the latest version KINARI-2 will demonstrate several current and future challenges.