Principle Investigator:  Stephen Power,
Department of Mathematics and Statistics,

Collaborators: 
Derek Kitson (Lancaster) EPSRC Research Associate from August 2012August 2014 Ghada Badri (Lancaster) James Cruickshank (University of Galway) Tony Nixon (Bristol University, Lancaster University) John Owen (Siemens, Cambridge) Bernd Schulze (Lancaster) 
Research Project summary (as stated in the Case for Support for EP/J008648/1):  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. 
Aims of the Project: 
(i) Determine the abstract rigid unit mode spectrum (RUM spectrum) and
mode multiplicity for crystal frameworks and semiinfinite crystal frameworks in dimensions 2, 3 and 4.
(ii) Develop operator theory and function theory methods in barjoint framework rigidity theory associated with general infinitesimal flexes, both for crystal frameworks and for general infinite frameworks. (iii) Determine characterisations of infinitesimal rigidity for barjoint frameworks constrained to surfaces both in the finite and infinite case, and obtain variants of FowlerGuest character formulae with respect to the crystallographic group. (iv) Obtain infinitesimal rigidity characterisations for infinite barjoint frameworks. In particular, determine CauchyEuler type results for the infinitesimal rigidity of infinitelyfaced convex deltahedra. 
Fulfillment of the Aims: 
These interrelated problem areas have been advanced in major ways. During the period of the grant there have been 14 published articles
related to these themes, authored or coauthored by the PI and the appointed RA, Dr Derek Kitson. These consist of 7 research journal publications (Outputs R1 to R7 below), 6 preprints placed on the mathematics archive (Outputs P8 to P13) and one conference proceedings (Output C1.) In addition there are several preprints in preparation
and the research is ongoing. (Dr Kitson now holds an indefinite Lectureship at Lancaster University.)
The PI and Dr Derek Kitson made significant progress in the general theory of infinite barjoint frameworks. In particular we obtained infinite framework generalisations of the fundamental combinatorial characterisations of Laman, for 2dimensional generic frameworks, and Tay, for bodybar frameworks (in arbitrary dimensions). Moreover we have broadened the general area of Geometric Rigidity in the direction of functional analysis by developing a nonEucldean metric theory for finite frameworks. In particular we obtained counterparts to the theorems of Laman and Tay as well as counterparts for the new infinite framework variants. These advances as well as other results in the problem areas (iii) and (iv) above are recorded in the Key Findings below for generic barjoint frameworks. Another deep highlight in the generic direction is the combinatorial characterisation of graphs that have isostatic placements on an algebraic surface of revolution. This is given Output R7, which is a continuation of the research direction of R1 with John Owen and Tony Nixon. The Key Findings for symmetric and crystallographic barjoint frameworks indicate advances in the general problem areas (i) and (ii). The topics of semiinfinite crystal frameworks and Toeplitz operator methods have so far been given less attention in favour of other functional analysis directions. These include a characterisation of almost periodic rigidity (a new but natural form of infinitesimal rigidity) and mathematical approaches to the RUM spectrum in crystallography including rigourous formulations of the RUM dimension. This work, as well as extensions to a mean flexibility dimension for quasicrystallographic frameworks, form part of ongoing research. 
Key Findings:  
I. Generic barjoint frameworks:  The key findings in the area of generic barjoint frameworks, with no assumptions of periodicity or symmetry, are the following.
1. A combinatorial characterisation of the infinitesimal rigidity of barjoint frameworks that are constrained to a cylinder in terms of (2,2)tight graphs. (Output R1.) 2. A combinatorial characterisation of the infinitesimal rigidity of barjoint frameworks that are constrained to an algebraic surface of revolution in terms of (2,1)tight graphs. (Output R7.) 3. A combinatorial characterisation of infinitesimal rigidity for barjoint frameworks in the nonEuclidean planes that are endowed with the classical pnorms in terms of (2,2)tight graphs. (Output R2.) 4. A generalisation of the fundamental combinatorial characterisation of Laman (1970) to countably infinite barjoint frameworks. (Output P8.) 5. A characterisation of the infinitesimal rigidity of countably infinite barjoint frameworks in terms of relatively rigid inclusion towers. (Output P8.) 6. A generalisation of the fundamental combinatorial characterisation of Tay (1984) to infinite bodybar frameworks, in all dimensions, both for the Euclidean and nonEuclidean norms. (Output P8.) 7. A generalisation of the Henneberg move constructibility of minimally rigid frameworks to infinite frameworks in the plane for both Euclidean and nonEuclidean norms. (Output P9.) 8. The determination of the infinitesimal flexibility dimension of infinitely faceted polyhedra. (Output P9.) 9. Combinatorial characterisations of infinitesimally rigid barjoint frameworks with respect to polyhedral norms. (Output P10.) 10. The formulation of the rigidity matrix for a barjoint framework in an arbitrary finitedimensional real normed space and the derivation of MaxwellLamantype counting conditions for infinitesimal rigidity. (Output P11.) 11. Combinatorial characterisations of block and hole graphs with (i) a single hole and multiple blocks, (ii) a single block and multiple holes, as well as a refutation of the more general conjecture of FinbowSingh and Whiteley (Isostatic Block and Hole Frameworks}, SIAM J. Discrete Math. 27 (2013) 9911020). (Output P13.) 12. The combinatorial characterisation of minimally rigid partial triangulations of the torus, the klein bottle and the real projective plane in the case of a single superficial hole. (Output P13.) 
II. Symmetric and crystallographic barjoint frameworks:  The main key findings for nongeneric barjoint frameworks are the following.
13. The derivation of symmetry adapted MaxwellLamantype counting conditions for infinitesimal rigidity in an arbitrary finitedimensional normed space. (Output P11.) 14. Rigourous definitions of the RUM spectrum and the RUM dimension of a crystal framework and its determination for key examples. (Outputs R3, R5.) 15. The introduction of Toeplitz operator methods for the infinitesimal flex analysis of bicrystals. (Output R4.) 16. A characterisation of the almost periodic rigidity of crystal frameworks in terms of the RUM spectrum. (Output R5. This includes a GALLERY of 10 diverse examples.) 17. A new derivation of the BorceaStreinu rigidity matrix for the affinely periodic flexibility of a crystal framework. (Output R6.) 18. The determination of general FowlerGuest symmetryadapted Maxwell Calladine formulae for crystal frameworks and general affinely periodic flexes. (Output R6.) 19. The combinatorial characterisations of minimally rigid 2dimensional barjoint frameworks with either reflectional or halfturn symmetry in the case of nonEuclidean norms with unit ball a quadrilateral. (Output P12.) 
Publications:  
(to September 2014):  Research Journal Publications:
R1. A. Nixon, J. C. Owen and S. C. Power, Rigidity of frameworks supported on surfaces, SIAM J. Discrete Math. 26 (2012), no. 4, 17331757. MR3022162, Zbl 1266.52018, arXiv:1009.3772, doi: 10.1137/110848852. R2. D. Kitson and S. C. Power, Infinitesimal rigidity for nonEuclidean barjoint frameworks, Bull. London Math. Soc. 46 (2014), 685697. doi:10.1112/blms/bdu017. R3. S. C. Power, Polynomials for crystal frameworks and the rigid unit mode spectrum, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), 27pp. no. 2008, 20120030, 27 pp. MR3158338, arXiv:1102.2744, doi: %10.1098/rsta.2012.0030. R4. S. C. Power, Crystal frameworks, matrixvalued functions and rigidity operators, Operator Theory: Advances and Applications. Volume 236 (2014), 405420. R5. G. Badri, D. Kitson and S. C. Power. The almost periodic rigidity of crystallographic barjoint frameworks, Symmetry 6 (2014), no. 2, 308328. arXiv:1402.6250. doi: 10.3390/sym6020308. R6. S. C. Power, Crystal frameworks, symmetry and affinely periodic flexes, New York J. Math. 20 (2014), 665693. R7. A. Nixon, J. C. Owen and S.C. Power, A characterisation of generically rigid frameworks on surfaces of revolution, to appear in the SIAM journal of Discrete and Applied Math. Preprints: P8. D. Kitson, S. C. Power, The rigidity of infinite graphs, 31 pages, submitted. This output is based on part of the 51 page article arXiv:1310.1860, 2013. P9. D. Kitson, S. C. Power, The rigidity of infinite graphs II: Inductive constructions. 20 pages, submitted. This output is also based on part of arXiv:1310.1860, 2013. P10. D. Kitson, Finite and infinitesimal rigidity with polyhedral norms, 26 pages, arXiv:1401.1336, 2014. P11. D. Kitson, B. Schulze, MaxwellLaman counts for isostatic barjoint frameworks in normed spaces, 17 pages, arXiv:1406.0998, 2014. P12. D. Kitson, B. Schulze, Rigidity characterisations for graphs with two edgedisjoint symmetric spanning trees, 21 pages, arXiv:1408.4637, 2014. P13. J. Cruickshank, D. Kitson and S. C. Power, The generic rigidity of triangulated surfaces with holes, to appear as an arXiv preprint, September/October 2014. Conference Proceedings: C1. S. D. Guest, P. W. Fowler and S.C. Power (Editors), Rigidity of periodic and symmetric structures in nature and engineering, Phil. Trans. R. Soc. A, vol 372, 2014. 
Disseminations:  
(to September 2014): 
Derek Kitson:
Fields Institute Toronto, Workshop on Making Models; stimulating research in Rigidity Theory and spatialvisual reasoning, Fields Institute Toronto, 59th August 2014. Lancaster University. Geometric Rigidity Workshop, 5th June 2014. Lancaster University, Pure Mathematics Seminar, 22nd May 2013. University of Glasgow, Analysis seminar, 14th May 2013. NUI Galway, Irish Geometry Conference, 9th May 2014. University of Liverpool, “Selected Topics in Mathematics” seminar, 7th March 2014. Oxford University, Functional Analysis seminar. 26th February 2013. Leeds University, Yorkshire Functional Analysis Group seminar, 16th October 2012. Lancaster University, Geometric Rigidity Workshop, 31th July 2013. Warsaw (WCMS). Workshop of the QOP network (Quantum groups, operators and noncommutative probability), 10th July 2013. Bristol University, Geometric and Topological Graph Theory workshop, 18th April 2013. Stephen Power: Fields Institute Toronto, Workshop on Making Models; stimulating research in Rigidity Theory and spatialvisual reasoning, Fields Institute Toronto, 59th August 2014. Lancaster University, Geometric Rigidity Workshop, 5th June 2014. University of Galway, 26th March, 2014. Trinity College Dublin (Dublin area Analysis Seminar), 25th March, 2014. Queen's University Belfast, Colloquium presentation, 21st March, 2014. Lancaster University, Geometric Rigidity Workshop, 31st July, 2013. Leeds University, QOP meeting, 24th September, 2013. Bristol University, Geometric and Topological Graph Theory workshop, 18th April 2013. Banff International Research Station workshop, {Rigidity Theory: Progress, Applications, and Key Problems,} July 2012. 