EPSRC Reference: 
EP/J008648/1 
Title: 
Crystal Frameworks, Operator Theory and Combinatorics 
Principal Investigator: 
Power, Professor S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics and Statistics 
Organisation: 
Lancaster University 
Scheme: 
Standard Research 
Starts: 
01 September 2012 
Ends: 
31 August 2014 
Value (£): 
158,111

EPSRC Research Topic Classifications: 
Logic & Combinatorics 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
A barjoint 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 (nondegenerate) foursided 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 barjoint framework is "typically" rigid in terms of its general shape, represented by the underlying edgevertex graph. The 2D problem was resolved by Laman in 1970, in terms of Maxwell counting conditions, and this introduced combinatorial (vertex/edge counting) tools and methods to go alongside the more obvious (but more cumbersome) method of simultaneous equation solving for vertex positions.
Infinite barjoint frameworks are not just a concern in pure mathematical analysis. They appear, for example, in mathematical models in Material Science, with the bars representing strong bonds in a crystal structure. For example, the basic silicate quartz, at different temperatures, provides a (topologically equivalent) pair of such frameworks, with the form of a periodic network of cornerlinked tetrahedra. The mathematics of lattice dynamics was developed in the first part of the last century, by Max Born and others, in order to understand the spectrum of phonon (vibration) modes in crystalline matter. However, for low energy modes 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.
The primary tool for understanding the rigidity of a finite framework structure is a matrix (an array of columns and rows of numbers) determined by the framework. This rigidity matrix, via linear algebra methods, detects the flexes of the structure. Also the implications of the symmetries of the structures can be detected and these associations remain true for infinite frameworks, although with added subtleties. The special case of periodic frameworks, with periodic flexibility and phaseperiodic flexibility, is a finitely determined context and the rigidity matrix gives rise to a finite matrix of multivariable polynomials. This function matrix gives a key new tool in the rigidity analysis of such frameworks.
In tandem with these methods, that go beyond simple linear algebra, there are also elaborations possible for the combinatorial tool of Maxwell counting. These refined methods take into account the symmetries of the structure and, in the crystal framework case, the symmetries of the crystallographic group.
Bondnode 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 will enrich such models through a systematic analysis of the infinitesimal dynamics and the flexibility of general barjoint structures, both finite and infinite.

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