Conference Agenda

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Session Overview
Session
MS175, part 1: Algebraic geometry and combinatorics of jammed structures
Time:
Wednesday, 10/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-111
30 seats, 56m^2

Presentations
3:00pm - 5:00pm

Algebraic geometry and combinatorics of jammed structures

Chair(s): Anthony Nixon (Lancaster), Louis Theran (St Andrews)

The minisymposium will combine the classical rigidity theory of linkages in discrete and computational geometry with the theory of circle packing, and patterns, on surfaces that arose from the study of 2- and 3-manifolds in geometry and topology. The aim being to facilitate interaction between these two areas. The classical theory of rigidity goes back to work by Euler and Cauchy on triangulated Euclidean polyhedra. The general area is concerned with the problem of determining the nature of the configuration space of geometric objects. In the modern theory the objects are geometric graphs (bar-joint structures) and the graph is rigid if the configuration space is finite (up to isometries). More generally one can consider tensegrity structures where distance constraints between points can be replaced by inequality constraints. The theory of (circle, disk and sphere) packings is vast and well known, with numerous practical applications. Of particular relevance here are conditions that result in the packing being non-deformable (jammed) as well as recent work on inversive distance packings. These inversive distance circle packings generalised the much studied tangency and overlapping packings by allowing ``adjacent'' circles to be disjoint, but with the control of an inversive distance parameter that measures the separation of the circles. The potential for overlap between these areas can be easily seen by modelling a packing of disks in the plane by a tensegrity structure where each disk is replaced by a point at its centre and the constraint that the disks cannot overlap becomes the constraint that the points cannot get closer together.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Flexibility of graphs on the sphere: the case of K_{3,3}

Matteo Gallet
JKU Linz

We present a study of necessary conditions for the edge lengths of minimally rigid graphs (Laman graphs) that make them mobile on the sphere. This is made possible by interpreting realizations of a graph on the sphere as elements of the moduli space of rational stable curves with marked points. By analyzing how curves of realizations intersect the boundary of this moduli space, we obtain a combinatorial characterization, in terms of colorings, for the existence of edge lengths that allow flexibility. We then give a classification of possible motions on the sphere of the bipartite graph with 3+3 vertices, for which no two vertices coincide.

This is a joint work with Georg Grasegger, Jan Legerský, and Josef Schicho.

 

Algebraic Geometry for Counting Realizations of Minimally Rigid Graphs

Georg Grasegger
JKU Linz

Minimally rigid graphs (Laman graphs) are defined to have only finitely many realizations in the euclidean plane, up to rotations and translations. It is known that the same graphs are also minimally rigid on the sphere. In this talk we present recent counting algorithms for this finite number of complex realizations both in the plane and on the sphere. Based on systems of polynomial equations we intrinsically use two different approaches from algebraic geometry to prove the algorithms. The necessary computations are, however, purely combinatorial and can be viewed in terms of graphs.

 

Pairing symmetry groups for spherical and Euclidean frameworks

Bernd Schulze
Lancaster

In this talk we will discuss the effect of symmetry on the infinitesimal rigidity of spherical frameworks and Euclidean bar-joint and point-hyperplane frameworks in general dimension. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for bar-joint frameworks with a set X of vertices collinear, spherical frameworks with vertices in X on the equator, and point-hyperplane frameworks with the vertices in X representing hyperplanes are all equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we can deduce some combinatorial consequences for the rigidity of both spherical and Euclidean frameworks.

This is joint work with Katie Clinch, Anthony Nixon and Walter Whiteley.