EPSRC Reference: 
EP/X012239/1 
Title: 
Exact Results in Aperiodic Systems 
Principal Investigator: 
Flicker, Dr F 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
School of Physics and Astronomy 
Organisation: 
Cardiff University 
Scheme: 
New Investigator Award 
Starts: 
01 April 2023 
Ends: 
30 September 2023 
Value (£): 
362,297

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Artificial Intelligence 
Condensed Matter Physics 
Logic & Combinatorics 
Magnetism/Magnetic Phenomena 
Surfaces & Interfaces 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
It is increasingly true that our modern world runs on solving problems of optimization subject to constraints. Examples include the scheduling of public transport, electrical power distribution, and the flow of internet traffic. Condensed matter physics  the study of that which emerges when huge numbers of particles interact  is an essential example: individual particles minimise their energies subject to the constraints of their mutual interactions and local environments. Understanding the resulting matter has enabled the technologies which run our lives.
An elegant description of this interplay of constraints and emergence is provided by dimer models. Dimers were originally conceived of as twoatom molecules landing on the surfaces of materials. How densely the dimers can pack onto the surface depends on the arrangement of surface atoms and the bonds connecting them. Understanding this process  adsorption  has never been more urgent. That is because it is an essential component in many carbon capture technologies (`dimers' in this case being carbon dioxide molecules, which contain three atoms lying in a straight line). Adsorption also forms the basis of catalysis, the process of reducing the energy required to carry out chemical reactions: if a molecule temporarily sticks to a surface the chance of it reacting with a second stuck molecule can be increased. Catalysis is used in the generation of over a trillion dollars' worth of products annually. The study of dimer models is therefore of pressing industrial and environmental concern.
The mathematical study of dimers has an exalted history within statistical physics, where it has led to some profound results. For example, 'conformal invariance' is a property appearing in diverse physical models from quantum gravity to water boiling under pressure  but the first formal proof of conformal invariance in statistical physics was found only recently, in a dimer model. Dimers were used to prove that there are exactly 12,988,816 ways to tile a chessboard with dominoes, and can be used to prove the following remarkable statement: if you deal a pack of cards into thirteen piles of four, it is always possible to choose one card from each pile so as to select one of each number.
Dr Flicker recently initiated the study of dimer models in a fundamentally new context: quasicrystals. The discovery of these remarkable materials in 1982 led to a Nobel prize, and forced a redefinition of what it means for something to be a crystal. Until then, all solids were believed to be either disordered on the atomic scale, or to have their atoms lined up in regular periodic structures, like the squares of a chessboard. Quasicrystals lack periodic structure, but nor are they disordered; instead they feature remarkable and beautiful symmetries: their structures can be described mathematically as slices through higherdimensional crystals. While many quasicrystals have now been grown artificially, only three naturally occurring quasicrystals have ever been found, all in the same Siberian meteorite.
Quasicrystals have potential advantages for adsorption: they are brittle, meaning they can be broken into particles only a few hundred atoms across, maximising their surface area; they remain solid at high temperatures, facilitating increased reaction rates; and their surface atoms feature a wide range of angles between neighbouring atomic bonds, which could be exploited by bendy molecules landing on these surfaces.
The aim of this project is to establish exact mathematical results in dimer models, and other statistical models of constrained optimization, on quasicrystals and related structures. Working with project partners  experimental physicists, machinelearning startups, and visual artists and composers  these results will be put to practical and industrial use, and will bring the inherent aesthetic appeal of these intriguing problems to a wider audience.

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Organisation Website: 
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