Gaps theorems and statistics of patterns in quasicrystals
Lead Research Organisation:
University of York
Department Name: Mathematics
Abstract
Much of the beauty in our universe arises in the emergence of order from complex systems. As scientists, our description of the natural world relies on our ability to describe this order. Symmetry is a valuable tool which sometimes allows us to simplify our description, but in truth many of the systems which we seek to describe are not perfectly symmetrical. This theme runs throughout the sciences and it also appears in many important problems in pure mathematics.
The research we are developing in this project will help us to understand patterns which come from a mathematical construction called the cut and project method. These patterns can be thought of as tilings of space. They are like the tilings that we see every day on walls, floors, and in artwork, except that they typically lack translational symmetry. Nevertheless, it is a fact that these patterns occur in the natural world, in viruses, in the study of energy states in quantum physics, and in recently discovered materials known as quasicrystals.
We will primarily be studying deformation properties and statistics of patterns in cut and project sets. This is a relatively new line of research, and our study will center around a connection which we have recently helped to develop, which involves a combination of ideas from the mathematical fields of number theory, topology, and dynamical systems.
To explain this connection in brief, to every `infinite' tiling of space we can associate a `finite' topological space. The topological space can be thought of conceptually as a donut, possibly with many (or even infinitely many) holes, with `fractal hair' growing out of every point on its surface. Even for mathematicians, this is a strange type of space, but we can understand something about it by using a tool from algebraic topology called cohomology. The cohomology of the topological space associated to a tiling is directly related to the complexity of the patterns which we see in the tiling. For example, if it turns out that our donut has two holes in it then the cohomology will detect this, and this in turn will tell us right away that the number of different configurations of tiles which we will see in our tiling is close to as small as theoretically possible. This connection also works the other way, which is to say that understanding patterns in the tiling also gives us information about the topology of the associated space. For cut and project sets the patterns in the tiling can be understood in terms of dynamical systems and number theoretic properties of the setup which produces them.
Our approach to these problems should help us to develop new mathematical methods to describe naturally occurring asymmetrical patterns. It is our hope that these methods will eventually find applications to real world problems in physics and biology.
The research we are developing in this project will help us to understand patterns which come from a mathematical construction called the cut and project method. These patterns can be thought of as tilings of space. They are like the tilings that we see every day on walls, floors, and in artwork, except that they typically lack translational symmetry. Nevertheless, it is a fact that these patterns occur in the natural world, in viruses, in the study of energy states in quantum physics, and in recently discovered materials known as quasicrystals.
We will primarily be studying deformation properties and statistics of patterns in cut and project sets. This is a relatively new line of research, and our study will center around a connection which we have recently helped to develop, which involves a combination of ideas from the mathematical fields of number theory, topology, and dynamical systems.
To explain this connection in brief, to every `infinite' tiling of space we can associate a `finite' topological space. The topological space can be thought of conceptually as a donut, possibly with many (or even infinitely many) holes, with `fractal hair' growing out of every point on its surface. Even for mathematicians, this is a strange type of space, but we can understand something about it by using a tool from algebraic topology called cohomology. The cohomology of the topological space associated to a tiling is directly related to the complexity of the patterns which we see in the tiling. For example, if it turns out that our donut has two holes in it then the cohomology will detect this, and this in turn will tell us right away that the number of different configurations of tiles which we will see in our tiling is close to as small as theoretically possible. This connection also works the other way, which is to say that understanding patterns in the tiling also gives us information about the topology of the associated space. For cut and project sets the patterns in the tiling can be understood in terms of dynamical systems and number theoretic properties of the setup which produces them.
Our approach to these problems should help us to develop new mathematical methods to describe naturally occurring asymmetrical patterns. It is our hope that these methods will eventually find applications to real world problems in physics and biology.
Planned Impact
The research in this proposal is mostly concentrated on intradisciplinary problems in mathematics. Most of the impact that it will generate will be realized first in mathematics and the rest of the sciences. However, as described in more detail in the Pathways to Impact attachment, some of the non-academic impacts could be:
1. A better understanding of the distribution of energy level spacing in the quantum harmonic oscillator.
2. A better understanding of certain metallic alloys and polymers.
3. Possible development of new techniques in virology.
4. Helping to maintain the international standing and image of the UK as a leader in science in technology.
1. A better understanding of the distribution of energy level spacing in the quantum harmonic oscillator.
2. A better understanding of certain metallic alloys and polymers.
3. Possible development of new techniques in virology.
4. Helping to maintain the international standing and image of the UK as a leader in science in technology.
Organisations
Publications
Beresnevich, V
(2017)
Sums of reciprocals of fractional parts and multiplicative Diophantine approximation
Järvenpää E
(2017)
Hitting probabilities of random covering sets in tori and metric spaces
in Electronic Journal of Probability
Haynes A
(2017)
Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II
in Indagationes Mathematicae
Haynes Alan
(2017)
Statistics of patterns in typical cut and project sets
in arXiv e-prints
Baake Michael
(2017)
A measure theoretic result for approximation by Delone sets
in arXiv e-prints
Haynes A
(2017)
A randomized version of the Littlewood Conjecture
in Journal of Number Theory
Gorodnik A
(2017)
Diophantine approximation for products of linear maps - logarithmic improvements
in Transactions of the American Mathematical Society
Walton J
(2017)
Pattern-equivariant homology
in Algebraic & Geometric Topology
Walton J
(2017)
Cohomology of rotational tiling spaces COHOMOLOGY OF ROTATIONAL TILING SPACES
in Bulletin of the London Mathematical Society
Haynes A
(2018)
Perfectly ordered quasicrystals and the Littlewood conjecture
in Transactions of the American Mathematical Society
Description | We have obtained a classification of all linearly repetitive cut and project sets, answering an open problem of Lagarias and Pleasants. We have also discovered how to reformulate the Littlewood Conjecture (a major unsolved problem in number theory) in terms of patterns in cut and project sets. Furthermore, (since the last submission Feb2017) we have proved the optimal random version of the conjecture. As a result of our findings, the members of our research team have been invited to an enormous number of conferences and universities, as invited, plenary, and colloquium level speakers. |
Exploitation Route | In the mathematical world, many people in the field of tiling theory are now trying to learn our methods. Several papers which use ideas related to those that we introduced to the subject have now emerged, and I suspect that there will be many more in the next few years. In the `real' world, our new results have direct implications for the structure of atoms in physical quasicrystals. There may also be implications in the field of virology, but we have not fully explored this outlet yet. |
Sectors | Digital/Communication/Information Technologies (including Software),Pharmaceuticals and Medical Biotechnology |
Description | Attendance and collaboration with researchers at conference in Lyon, France (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 4th-8th January 2016, attended conference "Dynamical Systems for Aperiodicity" in Lyon, France |
Year(s) Of Engagement Activity | 2016 |
URL | https://dsaperiodic2016.sciencesconf.org/ |
Description | Attended conference in Aarhus, Denmark, to collaborate with researchers in related fields (Jamie Walton) |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 13th-17th July 2015, attended conference 'Diophantine Approximation and Related Topics' |
Year(s) Of Engagement Activity | 2015 |
URL | http://mjcnt.phystech.edu/conference/aarhus/ |
Description | Colloquium talk at the University of St Andrews, Scotland: Henna Koivusalo |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Professional Practitioners |
Results and Impact | A talk on quasicrystals and their mathematical models to a general mathematical audience, informal discussions with the researchers at the institute, November 2015. |
Year(s) Of Engagement Activity | 2015 |
Description | Ergodic, Algebraic and Combinatorial Methods in Dimension Theory (February 15-19, 2016): Henna Koivusalo |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Workshop for researchers, focusing on recent advances at the interfaces of the above fields. Feb 15-19 2016. |
Year(s) Of Engagement Activity | 2016 |
URL | https://icerm.brown.edu/programs/sp-s16/w1/ |
Description | Invited for collaboration and to give talk at SDU, Odense (Jamie Walton). |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 24th January 2016, talk given and collaboration with researchers in SDU interested in my research. |
Year(s) Of Engagement Activity | 2016 |
URL | http://imada.sdu.dk/Research/GroupsAndAlgebras/ |
Description | Problem solving course (Univ York) |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | One of my postdocs (Henna Koivusalo) and I have been organizing and running a bi-weekly problem solving course at the University of York. The course is aimed at Y12 and Y13 students. The goals are: 1) To help them develop their mathematical reasoning and problem solving skills, 2) To give them the flavor of what it is to do mathematics, at the same time introducing them to more advanced and abstract material than they will learn in their A-levels, 3) To help the Y13 students prepare to take the STEP exam, to qualify for entry to and scholarships in the mathematics programs at top schools. |
Year(s) Of Engagement Activity | 2014,2015,2016 |
Description | Public lecture: London Mathematical Society Northern Regional Meeting: Henna Koivusalo |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Public/other audiences |
Results and Impact | As part of an LMS Regional meeting, a public talk on quasicrystals and their mathematical models, 23 June 2016. |
Year(s) Of Engagement Activity | 2016 |
URL | https://www.lms.ac.uk/content/lms-northern-regional-meeting |
Description | RI Masterclass (York) |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | I ran an RI Masterclass in York on the topics of large numbers and infinity. The audience were mostly Y9 students, and the goal was to give them a flavor of more advanced mathematical thinking, and to inspire some of them to want to purse further mathematical education. |
Year(s) Of Engagement Activity | 2016 |
Description | Research visit to Tel Aviv University, Israel, December 2015: Henna Koivusalo |
Form Of Engagement Activity | A formal working group, expert panel or dialogue |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Discussion with Proessor Barak Weiss (leading expert) on cut and project sets; seminar talk |
Year(s) Of Engagement Activity | 2015 |
Description | Summer school in tiling theory |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I was an invited speaker at a summer school on tiling theory, funded by CIRM, in Oleron, France. I gave a lecture series on repetitivity of patterns in mathematical models for quasicrystals. |
Year(s) Of Engagement Activity | 2016 |
URL | https://oleron.sciencesconf.org/ |
Description | Summer school on fractal geometry and complex dynamics |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a course at the Summer School on Fractal Geometry and Complex Dynamics, at Cal Poly San Luis Obispo. The course consisted of several lectures that described the basic theory of quasicrystals and their diffraction properties. |
Year(s) Of Engagement Activity | 2016 |
URL | http://www.calpoly.edu/~epearse/Fractals2016/ |
Description | Talk at California Polytechnic State University (Cal Poly), San Luis (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 22nd June 2016, talk given on recent research at Summer School on Fractal Geometry and Complex Dimensions |
Year(s) Of Engagement Activity | 2016 |
URL | http://www.calpoly.edu/~epearse/Fractals2016/timetable.html |
Description | Talk at University of Leicester (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 7th January 2015 Workshop on Aperiodic order, talk given on current research. |
Year(s) Of Engagement Activity | 2008,2015 |
URL | http://www2.le.ac.uk/departments/mathematics/research/topological/related-events-1/copy_of_related-e... |
Description | Talk given and attendance at "Arbeitsgemeinschaft", a workshop at MFO in Oberwolfach, Germany (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 4th-9th October 2015, collaboration and talk given at the workshop: "Arbeitsgemeinschaft: Mathematical Quasicrystals" |
Year(s) Of Engagement Activity | 2015 |
URL | https://www.mfo.de/occasion/1541/www_view |
Description | Talk in NUI, Galway (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 25th June 2015, talk given on recent research at the 30th Summer Conference on Topology & its Applications. |
Year(s) Of Engagement Activity | 2015 |
URL | http://www.conference.ie/Conferences/index.asp?Conference=393 |
Description | Talk in San Antonio (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Industry/Business |
Results and Impact | 13th January 2015, Joint Mathematics Meetings, talk given on recent research. |
Year(s) Of Engagement Activity | 2015 |
URL | http://jointmathematicsmeetings.org/meetings/national/jmm2015/2168_intro |
Description | Talk in pure mathematics seminar, at University of Leicester (Jamie Walton) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Industry/Business |
Results and Impact | 9th December 2014. Talk on result results from research on the grant. |
Year(s) Of Engagement Activity | 2014 |
URL | http://www2.le.ac.uk/departments/mathematics/research/pure/colloquium |
Description | Talk on cut and project sets at British mathematical Collquium 21-24 March 2016: Henna Koivusalo |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Professional Practitioners |
Results and Impact | Talk for a specialist audience 22 March 2016 |
Year(s) Of Engagement Activity | 2016 |
URL | https://people.maths.bris.ac.uk/~matyd/BMC/schedule.html |
Description | University of York Undergraduate Mathematics Society (MathSoc) talk: Extreme complexity of fractal sets: Henna Koivusalo |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Public/other audiences |
Results and Impact | Public talk advertised and aimed to undergraduates and general public |
Year(s) Of Engagement Activity | 2015 |
URL | http://yums.org.uk/2015/03/extreme-complexity-of-fractals-sets-by-henna-koivusalo/ |