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## Discrete Mathematics and Geometric Rigidity

### Organisation profile

Geometric Rigidity theory is concerned with the rigidity of structures which are defined by geometric constraints (fixed lengths, fixed areas, fixed directions, etc.) on a set of rigid objects (points, line segments, polygons, etc.). The origins reach back to 1813 and Cauchy's rigidity theorem for a convex triangulated polyhedron with rigid edges and flexible joints. The theory is also concerned with the infinitesimal flexibility and continuous flexibility of structures, ranging from Victorian mechanisms, to robot arms and the crystal networks of zeolites.

‌The mathematics of rigidity has both combinatorial and geometric aspects and draws on techniques from a wide range of mathematical areas, including graph theory, matroid theory, positive semidefinite programming, representation theory of symmetry groups and algebraic and projective geometry. When the structures are infinite, as in a mathematical model for a periodic crystal for example, methods from functional analysis and operator theory become relevant.

The rigidity and flexibility properties of a structure — either man-made, such as a building, bridge or mechanical linkage, or found in nature, such as a biomolecule, protein or crystal — are critical to its form, behaviour and functioning. Thus rigidity theory has many practical applications in fields such as engineering, robotics, materials science, medicine and biochemistry. The methodology of rigidity and configuration uniqueness also has applications in computer-aided design, network sensoring and NMR reconstruction.

The mathematical theory embraces on the one hand the analysis of generic structures, in which the combinatorics of connectivity play the dominant role, and, on the other hand, structures with symmetries and regularities where symmetry group actions are significant. Indeed the rigidity analysis of symmetric structures is an important area of research which has seen a dramatic increase in interest over the last few years, both in mathematics and in the applied sciences.

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• Forthcoming

## Crystal flex bases and the RUM spectrum

Power, S., Kitson, D. & Badri, G., 31/05/2021, (Accepted/In press) In: Proceedings of the Edinburgh Mathematical Society.

Research output: Contribution to journalJournal articlepeer-review

• Forthcoming

## The first-order flexibility of a crystallographic framework

Power, S. & Kastis, E., 28/05/2021, (Accepted/In press) In: Journal of Mathematical Analysis and Applications. 16 p.

Research output: Contribution to journalJournal articlepeer-review

• Published

## Symbol functions for symmetric frameworks

Kastis, E., Kitson, D. & McCarthy, J., 15/05/2021, In: Journal of Mathematical Analysis and Applications. 497, 2, 24 p., 124895.

Research output: Contribution to journalJournal articlepeer-review

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• ## American Institute of Mathematics: Rigidity and flexibility of microstructures

Activity: Participating in or organising an event typesParticipation in workshop, seminar, course

• ## Galway Mathematics Seminar

Activity: Talk or presentation typesInvited talk

• ## SIAM conference on applied algebraic geometry

Activity: Participating in or organising an event typesParticipation in conference