Other speakers
Senja Dominique Barthel (EPFL)
Predicting braids for coordination polymers
Coordination polymers (CP) consist of central nodes (usual metals) that are connected by organic or inorganic linkers to form a material that extends periodically in 1, 2 or 3 directions. Here we focus on CPs that extend periodically in one direction, consisting of interwoven strands. We show how the theory of mathematical braids can help to model possible molecular braid formations. This is joint work with Davide M. Proserpio and Igor Baburin.
Evangelos Bartzos (University of Athens), Georg Grasegger (RICAM / JKU) and Jan Legersky (RISC, JKU)
Lower bounds on the maximal number of realizations
We present recent results on bounds for the number of realizations of minimally rigid graphs in plane and space. We consider complex solutions of the corresponding algebraic systems and give asymptotic bounds for the maximal number of realizations. Furthermore, we introduce a method for specifying edge lengths allowing many real solutions.
Ciprian Borcea (Rider University)
New pathways for auxetics
Auxetic behaviour refers to lateral widening upon stretching. For auxetic periodic frameworks, a strictly geometric approach leads to solutions of the twin problems of structure and design. We report on these new results and illustrate the same design procedures. This is joint work with Ileana Streinu.
Jim Cruickshank (NUI Galway)
On (2,2)-tight torus graphs of genus one
We will discuss an inductive construction of (2,2)-tight graphs that can be embedded in a torus without edge crossings. Our construction uses certain splitting moves which preserve the topology of the underlying surface. These splitting moves, which are topologically dual to Henneberg moves, are natural to consider in the context of certain geometric applications. This is joint work with Derek Kitson, Steve Power and Qays Shakir.
Sean Dewar (Lancaster University)
Framework rigidity in normed spaces
In 1978 L. Asimow and B. Roth proved that a regular framework could be deformed by an edge-length preserving motion of its vertices if and only if it could be deformed by an edge-length preserving infinitesimal motion of its vertices. We give an extension of the main theorem from L. Asimow and B. Roth's 1978 paper for any finite-dimensional normed space. This equivalence shall be shown to hold for almost all placements of a given graph. We shall further give necessary conditions for a graph to have a rigid placement in a given normed space and a sufficient condition if the graph is small relative to the dimension of the normed space.
Georg Grasegger (RICAM / JKU), Jan Legersky (RISC, JKU) and Josef Schicho (RISC, JKU)
Flexible labellings of rigid graphs
Given a graph, we ask whether it is possible to find a flexible labelling, namely, edge lengths such that there are infinitely many compatible realizations, modulo rigid motions. Even if a graph is generically rigid, then non-generic edge lengths may still be flexible. A classical construction by A. Dixon from 1899 works for all bipartite graphs. In this talk, we give a combinatorial criterion for the existence of a flexible labelling.
Zeyuan He (University of Cambridge)
A theoretical framework for rigid origami - He
Rigid origami is usually modelled to fold and unfold a paper by bending along creases without deformation of the panels, which is a new branch of the classic rigidity theory. Here we will introduce a theoretical framework to discuss rigid origami, present some new conclusions, and review some important previous results under this framework. We start with general definitions of concepts in rigid origami, then focus on the configuration space and rigid-foldability for a given creased paper. Finally, we will show some families of rigid-foldable creased papers.
Reference: Zeyuan He and Simon D. Guest, On rigid origami I: piecewise-planar paper with straight-line creases, arXiv:1803.01430, 2018.
Bill Jackson (Queen Mary University of London)
The rigidity of linearly constrained frameworks
We consider the problem of characterising the generic rigidity of bar-joint frameworks in R^d in which each vertex is constrained to lie in a given affine subspace. The special case when d=2 was previously solved by Ileana Streinu and Louis Theran in 2010. We will extend their characterisation to the case when d\geq 3 and each vertex is constrained to lie in an affine subspace of dimension t, when t=1,2 and also when t\geq 3 and d\geq t(t-1). This is joint work with James Cruickshank, Hakan Guler and Anthony Nixon.
Tibor Jordan (Eotvos Lorand University)
Compressed frameworks and compressible graphs
We consider d-dimensional bar-and-joint frameworks (G,p). We say that a framework (G,p) is compressed if for every equivalent framework (G,q) and for every vertex pair u,v the u-v-distance in (G,q) is not less than the u-v-distance in (G,p). A graph is said to be compressible in d-space if it has a compressed d-dimensional realization (G,p). These notions will be studied in this talk, including some partial results. (Joint work with Jialin Zhang.)
Eleftherios Kastis (Lancaster University)
The first-order flexibility of a crystal framework
For a general crystal framework, the space of all complex infinitesimal flexes is the vector space of C^d-valued velocity fields on the joints of the framework that satisfy the first-order flex condition for every bar. Due to the periodic structure of the framework the flex space is invariant under the natural translation operators and is closed with respect to the topology of coordinatewise convergence. In this talk, we shall indicate some new methods from analysis and commutative algebra. In particular, we generalise Lefranc's spectral synthesis theorem for the case r = 1 to a vector-valued setting, to obtain that the flex space of a crystal framework is the closed linear span of flexes which are vector-valued polynomially weighted geometric multi-sequences (Kastis & Power, 2018).
Csaba Kiraly (MTA-ELTE Egervary Research Group on Combinatorial Optimization and Eotvos Lorand University)
Packing arborescences with matroid constraints via matroid intersection
Katoh and Tanigawa characterized the rigidity of slider-pin frameworks. Their paper on the combinatorial details of this rigidity matroid initiated intensive research in combinatorial optimization on packing arborescences with matroid constraints. Here a packing means edge-disjoint subgraphs. The first such result was due to Durand de Gevigney, Nguyen and Szigeti that gave an alternative proof for one of the main results of a paper by Katoh and Tanigawa through an arborescence packing with a matroid constraint and an orientation result. This packing theorem generalized Edmonds' fundamental theorem on packing spanning arborescences. Later common generalizations of this matroid constrained packing result and other previous generalizations of Edmonds' theorem were discovered.
It was also discovered by Edmonds that the original problem of packing spanning arborescences can be characterized through the matroid intersection theorem of Edmonds which also implies an efficient algorithm for the weighted case of the problem. In my talk, I show how a similar characterization can be given for the most recent results on arborescence packings. To define the matroid in the characterization, I also show how a new class of matroids can be defined by extending an earlier construction of matroids from intersecting submodular functions to bi-set functions based on an idea of Frank. Joint work with Zoltán Zsigeti and Shin-ichi Tanigawa.
John Owen (Siemens)
Constraints between linear subspaces in 2 and 3 dimensions
I will discuss results and conjectures for the generic rigidity matroid which is determined by distance and angle constraints between points, lines and planes in 2 and 3 dimensions.
Hattie Serocold (Lancaster University)
Isostatic frameworks with classes of coordinated bars
We model coordinated frameworks using graphs with an edge colouring - motions of the framework (G,c,p) are permitted to increase or decrease the lengths of these coloured edges but must have the same effect on all edges within a colour class. Uncoloured edges are considered as bars with length fixed by the embedding p, as in the case of standard frameworks (G,p). We shall extend standard characterisations of infinitesimally rigid uncoloured graphs to our coordinated context, in the case of one and two classes of coordinated edges.
Qays Shakir (NUI Galway)
Contact graphs of circular arcs
We will discuss representations of surface graphs as contact graphs of configurations of circular arcs. In this representation, the vertices of the graphs are represented by circular arcs in surfaces of constant curvature while their edges are represented by the contacts of circular arcs. We first review some previously known results for contact representations in the plane. Then we show that every (2,2)-tight torus graph can be represented by a circular arc configuration in the flat torus. This work forms part of a joint project with James Cruickshank, Derek Kitson and Stephen Power.
Meera Sitharam (University of Florida)
Fast Flip Realizations and Flex Contact Paths of Planar Isostatic Linkages
Although deciding existence of a 2D realization of an isostatic linkage with planar underlying graphs is NP-hard, we show that each realization has a unique index called a "flip," and given a flip, finding the corresponding realization, if it exists, has a polynomial time algorithm in a standard finite precision computational model.
The key result is an inductive construction (in fact at least 3) that leverage the combination of planarity and isostaticity of the graph: the graph is constructed one vertex at a time, never deleting edges, and maintaining at most 3 degrees of freedom at every step. Furthermore, it is possible to remove an edge at each step of the construction, replacing it with a non-edge and yielding a 2-tree. The algorithm searches the space of non-edge lengths to find the realization with the given flip. We have implemented this algorithm and are able to find flip realizations for quite large linkages (hundred vertices) in real time. We are now working towards flip realizations for linkages with thousands of vertices. If time permits, I will mention initial explorations into related questions on so-called flex paths and flex contact paths between isolated 2D configurations of a planar isostatic linkage.
A flex path is a path in the union of configuration spaces of the linkages minus some single edge; and flex contact paths are between isolated configurations of a unit distance planar linkage, i.e. a path in the union of configuration spaces of all 1-dof unit distance mechanisms. The question is interesting even without planarity and in higher dimensions. We have implemented this algorithm for the unit distance linkages in 3D with additional noncolliding conditions.
The work is joint with my students Mingyu Kang, Jeremy Youngquist and Rahul Prabhu, and arises from discussions on glasses and colloids with Bob Connelly, Shlomo Gortler, Miranda Holmes-Cerfon, Louis Theran and Mike Thorpe. Different aspects of the work are supported by 2 grants from NSF-DMS and a grant from DARPA.
Louis Theran (University of St Andrews)
Unlabelled rigidity problems
A framework (G,p) is globally rigid in dimension d, if, given the edge length measurements v := m_G(p) one can recover p, up to an unknown isometry, from G and v. The unlabelled version of this problem asks when we can recover p from only the measurement set v (and no association to a graph). When p is generic and the number of vertices is known, a sufficient condition on G for unlabelled reconstruction is that it is generically globally rigid and has at least d + 2 vertices. This extends a result of Boutin and Kemper, who showed that unlabelled reconstruction is possible when G is a complete graph. Joint work with Shlomo Gortler and Dylan Thurston.
