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Crystal Frameworks, Operator Theory and Combinatorics

Project: Research

Description

A bar-joint framework is the mathematical abstraction of a structure in space formed by connecting stiff bars at their endpoints by joints with arbitrary flexibility. The structure may be flexible, as with a (non-degenerate) four-sided framework F in the plane, or may be rigid, as occurs when a diagonal bar is added to F. A subtle open problem is to characterise when a 3D bar-joint framework is "typically" rigid in terms of its general shape, represented by the underlying edge-vertex graph. The 2D problem was resolved by Laman in 1970, in terms of Maxwell counting conditions, and this introduced combinatorial and matroid tools and methods to go alongside the more obvious (but more cumbersome) method of simultaneous equation solving for vertex positions.

Infinite bar-joint frameworks also appear as bond-node models in Material Science, with the bars representing strong bonds in a crystal structure. Silicate quartz, for example, at different temperatures gives rise to topologically equivalent geometrically distinct periodic frameworks of corner-linked tetrahedra. The mathematics of lattice dynamics was developed by physicists in order to understand the spectrum of phonon excitation modes in crystalline matter. However, for low energy modes (RUMs) it turns out that there is a (less cumbersome) approach based on the infinitesimal rigidity theory of infinite frameworks and this is one of the themes of the research project.

Bond-node structures are ubiquitous in mathematical models in Material Science (mathematical quasicrystals, for example), in Engineering (space structures, for example) and in computer aided design (in the mathematics of sequentially constructed CAD diagrams, for example). The research project aims to enrich such models through a systematic analysis of the infinitesimal dynamics, rigidity and flexibility of general bar-joint structures, both finite and infinite. Also the mathematical analysis aims to establish new fundamental combinatorial characterisations and new methods drawn from functional analysis perspectives.
AcronymCrystal Frameworks, Operator Theory and Combinatorics
StatusFinished
Effective start/end date1/09/1231/08/14

Research outputs