Cluster algebras with periodicity and discrete dynamics over finite fields
Lead Research Organisation:
University of Kent
Department Name: Sch of Maths Statistics & Actuarial Sci
Abstract
This project is about bringing together ideas from three different areas of mathematics, namely algebra, mathematical physics, and number theory. It begins by considering some new objects in algebra, called cluster algebras, which have been around for just over a decade, and now constitute the most rapidly developing part of the subject. The project then looks within cluster algebras to find examples of discrete integrable systems, which are used by physicists to describe interactions between particles which evolve in discrete time steps. It then goes on to examine this evolution in discrete time from the viewpoint of number theory - dynamics over finite fields, to be precise - and ends up by showing how to generate number sequences with random behaviour from this discrete dynamics.
Some more details of the three different parts of the project are explained below.
I) Cluster algebras:
Most structures in algebra are defined by giving a set of objects, or generators, and then providing a set of relations or equations that they satisfy, which fix the rules for combining them. Cluster algebras are rather different, in that the generators are not given from the start, but are constructed in bunches (called clusters) from an initial set (called a seed) via a recursive process called mutation. The process of mutation is very complicated: at each step there are different paths to choose from, and generally the rules of mutation get changed along the way.
The inherent complexity of mutation is the main reason why the general structure of cluster algebras is not very well understood. However, in certain cases, the cluster algebra can end up looking the same after a finite number of mutation steps, up to symmetry. This periodic behaviour under sequences of mutations means that clusters can be produced by means of a single dynamical system, that is, a system which evolves in discrete time steps. The first main aim of the project is to classify the situations in which this periodicity arises.
II) Discrete integrable systems:
The laws of physics, such as Newton's laws of motion, are traditionally described mathematically in terms of differential equations, which describe how certain quantities vary smoothly over space and time. It was realized towards the end of the 19th century that, as soon as more than two particles interact, it is generally impossible to solve (or integrate) the laws of motion, and the behaviour of the system may even be chaotic. Nevertheless, exactly solvable (or integrable) systems helped to develop quantum theory, and enjoyed a renaissance in the second half of the 20th century when they were found in the description of stable solitary waves on shallow water, known as solitons.
The advent of soliton theory led to increased efforts to extend the properties of integrability to the discrete domain - both in space (evolution on a lattice), and in time (dynamics in discrete steps) - to produce discrete integrable systems. Very recently, I have found new examples of discrete integrable systems that appear inside cluster algebras with periodicity, and it is another main aim of my work to understand precisely when this happens.
III) Discrete dynamics over finite fields:
Imagine an unusual clock which is labelled with the numbers 1 up to p, where (unlike the number 12) p is a prime number, and in each hour the short hand moves from one number to the next. Starting from a given "time", the short hand will return to the same number every p hours. Thus one can add the numbers on the clock, forgetting any extra multiples of p. It turns out that one can also multiply, as well as subtract and divide (except by p), so that this "clock arithmetic" is a number system with p elements: the simplest example of a finite field.
I aim to show that, over finite fields, discrete integrable systems from cluster algebras give number sequences with exactly the properties required to make codes.
Some more details of the three different parts of the project are explained below.
I) Cluster algebras:
Most structures in algebra are defined by giving a set of objects, or generators, and then providing a set of relations or equations that they satisfy, which fix the rules for combining them. Cluster algebras are rather different, in that the generators are not given from the start, but are constructed in bunches (called clusters) from an initial set (called a seed) via a recursive process called mutation. The process of mutation is very complicated: at each step there are different paths to choose from, and generally the rules of mutation get changed along the way.
The inherent complexity of mutation is the main reason why the general structure of cluster algebras is not very well understood. However, in certain cases, the cluster algebra can end up looking the same after a finite number of mutation steps, up to symmetry. This periodic behaviour under sequences of mutations means that clusters can be produced by means of a single dynamical system, that is, a system which evolves in discrete time steps. The first main aim of the project is to classify the situations in which this periodicity arises.
II) Discrete integrable systems:
The laws of physics, such as Newton's laws of motion, are traditionally described mathematically in terms of differential equations, which describe how certain quantities vary smoothly over space and time. It was realized towards the end of the 19th century that, as soon as more than two particles interact, it is generally impossible to solve (or integrate) the laws of motion, and the behaviour of the system may even be chaotic. Nevertheless, exactly solvable (or integrable) systems helped to develop quantum theory, and enjoyed a renaissance in the second half of the 20th century when they were found in the description of stable solitary waves on shallow water, known as solitons.
The advent of soliton theory led to increased efforts to extend the properties of integrability to the discrete domain - both in space (evolution on a lattice), and in time (dynamics in discrete steps) - to produce discrete integrable systems. Very recently, I have found new examples of discrete integrable systems that appear inside cluster algebras with periodicity, and it is another main aim of my work to understand precisely when this happens.
III) Discrete dynamics over finite fields:
Imagine an unusual clock which is labelled with the numbers 1 up to p, where (unlike the number 12) p is a prime number, and in each hour the short hand moves from one number to the next. Starting from a given "time", the short hand will return to the same number every p hours. Thus one can add the numbers on the clock, forgetting any extra multiples of p. It turns out that one can also multiply, as well as subtract and divide (except by p), so that this "clock arithmetic" is a number system with p elements: the simplest example of a finite field.
I aim to show that, over finite fields, discrete integrable systems from cluster algebras give number sequences with exactly the properties required to make codes.
Planned Impact
The direct impact of this research will be to academic beneficiaries, as described above. However, one of the main conclusions of the 2012 Deloitte report commissioned by EPSRC was that mathematical research is a driver of long-term economic growth, so in the longer term this project should be of broad benefit to the UK economy.
It is very hard to predict exactly how a particular piece of mathematical research will make a long-term impact on society, but for this project there is one particular area that I am going to explore, namely the potential use of the discrete dynamical systems being considered as pseudorandom number generators (PRNGs). Many applications, such as coding (e.g. for efficient storage and transmission of data), cryptography (e.g. for secure web transactions) or Monte Carlo methods (e.g. for numerical simulations in finance) require a source of random numbers. However, true random numbers (as generated by a physical process like dice rolling, or thermal noise) are costly and time-consuming to generate. This is why PRNGs, which are obtained from a deterministic process, but have approximately the same statistical properties as random numbers, are more commonly used in applications. In due course I will make links with users of PRNGs by presenting my results at international meetings on computational number theory and cryptography. Since such meetings are often attended by researchers from industry, this will allow me to assess the potential practical applications of my results.
Having said that, I can predict that the main immediate impact of the project beyond academia will be through the extensive programme of outreach activities that I am planning. I will advertise my research to the public through workshops and Royal Institution Mathematics Masterclasses for school children, continuing professional development activities for school mathematics teachers, Nuffield Foundation summer projects and presentations at national science fairs. (For a full description of my planned activities for outreach, as well as my strong track record in this area, please read the Pathways to Impact statement.)
I am passionate about communicating mathematics, and I hope that my own enthusiasm for the subject will inspire the next generation of mathematicians in the UK.
It is very hard to predict exactly how a particular piece of mathematical research will make a long-term impact on society, but for this project there is one particular area that I am going to explore, namely the potential use of the discrete dynamical systems being considered as pseudorandom number generators (PRNGs). Many applications, such as coding (e.g. for efficient storage and transmission of data), cryptography (e.g. for secure web transactions) or Monte Carlo methods (e.g. for numerical simulations in finance) require a source of random numbers. However, true random numbers (as generated by a physical process like dice rolling, or thermal noise) are costly and time-consuming to generate. This is why PRNGs, which are obtained from a deterministic process, but have approximately the same statistical properties as random numbers, are more commonly used in applications. In due course I will make links with users of PRNGs by presenting my results at international meetings on computational number theory and cryptography. Since such meetings are often attended by researchers from industry, this will allow me to assess the potential practical applications of my results.
Having said that, I can predict that the main immediate impact of the project beyond academia will be through the extensive programme of outreach activities that I am planning. I will advertise my research to the public through workshops and Royal Institution Mathematics Masterclasses for school children, continuing professional development activities for school mathematics teachers, Nuffield Foundation summer projects and presentations at national science fairs. (For a full description of my planned activities for outreach, as well as my strong track record in this area, please read the Pathways to Impact statement.)
I am passionate about communicating mathematics, and I hope that my own enthusiasm for the subject will inspire the next generation of mathematicians in the UK.
Organisations
- University of Kent (Fellow, Lead Research Organisation)
- University of Tokyo (Collaboration)
- Chinese Academy of Sciences (Collaboration)
- China University of Mining and Technology (CUMT) (Collaboration)
- Roma Tre University (Collaboration)
- UNSW Sydney (Collaboration)
- University of Brescia (Collaboration)
- University of La Rioja (Collaboration)
- Polytechnic University of Catalonia (Collaboration)
- Tsinghua University China (Collaboration)
- La Trobe University (Collaboration)
People |
ORCID iD |
| Andrew Hone (Principal Investigator / Fellow) |
Publications
Andrew N. W. Hone
(2018)
On a Family of Sequences Related to Chebyshev Polynomials
in Journal of Integer Sequences
Barnes L
(2019)
Dynamics of conservative peakons in a system of Popowicz
in Physics Letters A
Barnes L
(2022)
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in ????????????? ? ?????????????? ??????
Barnes L
(2022)
Similarity reductions of peakon equations: integrable cubic equations
in Journal of Physics A: Mathematical and Theoretical
Fedorov Y
(2015)
Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties
in Journal of Integrable Systems
Felikson A
(2023)
Exchange graphs for mutation-finite non-integer quivers
in Journal of Geometry and Physics
Garcia Elsener A
(2019)
Factoriality and class groups of cluster algebras
in Advances in Mathematics
Hamad K
(2018)
QRT maps and related Laurent systems
in Advances in Applied Mathematics
Hone A
(2015)
Continued fractions for some transcendental numbers
in Monatshefte für Mathematik
Hone A
(2016)
On the continued fraction expansion of certain Engel series
in Journal of Number Theory
| Description | When hours are measured on a clock, the time repeats every 12 hours: this is called clock arithmetic (or modular arithmetic). If the number of hours is a prime number, then clock arithmetic produces a finite field. Cluster algebras have remained a hot topic in modern algebra since they were introduced by Fomin & Zelevinsky around the year 2000 - the name stems from the fact that the generating elements are naturally grouped into sets of fixed size, called clusters, produced by a recursive process called mutation. When the cluster algebra displays periodic behaviour, certain sequences of mutations correspond to a discrete dynamical system, where time varies in jumps rather than continuously. Using ideas from physics, we have classified families of cluster algebras and found many new generalizations (LP algebras) with discrete dynamics that conserves quantities analogous to the energy of a system of particles. Moreover, we have used finite field dynamics as a tool to identify when this happens, and connected some of these systems with new types of sums represented as certain infinite fractions (continued fractions) arising in number theory. Our results on explicit continued fraction expansions were developed in a different direction by a researcher in Spain (Varona), with whom we subsequently developed a new collaboration, and more recently they have been further extended and generalized by a group of mathematicians from Japan and France. The work that we have done on generalized cluster structures for difference equations (discrete dynamics) has sparked a new collaboration with a group of researchers in Tokyo, with two years' funding from the Royal Society. Currently we are still exploring applications of these results over finite fields, or more general modular arithmetic (residue rings) to develop computer algorithms for factoring numbers, encrypting messages, and sequences of numbers that look random (pseudorandom number generation). Some of our original objectives in this area were very ambitious, and there is a lot more work to be done to fully explore the ramifications of what we have found so far. Increased research capability in mathematics has arisen from the training of two postdoctoral RAs and one PhD student. One of the RAs left early to take up a permanent lecturing position in Flensburg, Germany, while the other is now a visiting researcher at Lincoln, and the former PhD student has just started a JSPS Postdoctoral Fellowship in Chiba, Japan. Additional capability was produced by the fact that the majority of the PI's teaching was bought out by the EPSRC Fellowship, which allowed Kent to hire a Temporary Lecturer in Applied Mathematics (Xenitidis) to cover this teaching; Xenitidis is a highly active researcher who formerly held a Newton Fellowship, and moved on to a position as Senior Lecturer at Liverpool Hope University. Due to being adversely affected by the coronavirus pandemic. the project has been extended after its end date for 7 months into 2021, with extra funding from UKRI via the University of Kent's UKRI Covid-19 Grant Extension Allocation fund, which should provide extra time to write up and disseminate some significant additional results from the project. Hence I have answered below that the award is still active, even though the official end date of the project has passed. Nevertheless, despite the fact that several results remain to be written up, and many open problems remain, a vast proportion of our objectives and aims concerning cluster algebras and higher-dimensional dynamics over finite fields have been met. Some problems were completely solved by members of our team (e.g. Lax pairs and cluster structures of discrete KP reductions, construction of periodic LP algebras), while other problems (e.g. classifying cluster algebras with periodicity, and entropy of the associated discrete dynamics) were being considered simultaneously by groups elsewhere in the world, who came up with their own solutions during the lifetime of the project, so these objectives were effectively closed off. The project has also produced unexpected ideas that will continue to be a source of inspiration for many years to come, |
| Exploitation Route | We presented some of the outcomes of the project at conferences on computational intelligence and symbolic computation, which has led to discussions with computer science practitioners working in cybersecurity, and visits to Google. We plan to take these results forward by additional funding applications to develop these ideas. Through our public engagement activities throughout the project we have also identified certain outcomes that would be highly suitable for a citizen science project, particularly involving school students, and we have found partners willing to support/sponsor such a project (the Institute for Research in Schools https://researchinschools.org/, Maplesoft https://maplesoft.com/). Again, we are looking into additional funding for this. |
| Sectors | Digital/Communication/Information Technologies (including Software),Education,Security and Diplomacy |
| Description | In November 2015 I participated in What's Your Angle at the Science Museum, the LMS 150th Anniversary Festival. Together with a group of researchers from Kent, our team presented research on solitons and integrable systems to groups of schoolchildren and other members of the public. This outreach activity was not directly linked to findings from my current project (which had only just started), but nevertheless it was closely related to the context of my research in mathematical physics. Moreover, it gave me ideas about ways to present my current ideas to the public in the near future. I have given several presentations in schools, both in the UK and in Australia while a Visiting Professorial Fellow at UNSW, Sydney and I am developing project material on discrete dynamics and finite fields. This will be made available online by the Institute for Research in Schools (IRIS - see https://researchinschools.org/), so that students in schools throughout the UK can do project work on topics related to the project. In late 2017 I became involved in a project through the Seqfan mailing list associated with the Online Encyclopedia of Integer Sequences (OEIS). It concerned prime values in sequences of Chebyshev polynomials evaluated at integer arguments. At the suggestion of Neil Sloane (the originator of OEIS) I began to work on the project with a group of collaborators who were recreational mathematicians, and ended up publishing a paper with two of them (Jeffery and Selcoe) in Journal of Integer Sequences in 2018. During the first lockdown in 2020 I began further conversations with IRIS about developing some of the findings into a project for school students (informal title: "Huge Primes") to engage them with research on number theory and cryptography, and Maplesoft has agreed to provide sponsorship in the form of free software licenses for participating students. In early 2022, I was awarded a small amount of support from Kent, as part of its allocation of Research England Participatory Research funding. This covered some initial costs for IRIS to prepare some of the online material for the Huge Primes project, as well the cost of briefly employing a part-time as a research assistant at Kent to develop some of the relevant content on cryptography in a form suitable for schools. (We employed Joe Mist, a student in Computing at Kent taking the MSc in Cyber Security.) We are currently seeking additional funding to support this: in particular we are considering whether Hone should apply for a KTP with IRIS. In the meantime, we have started a pilot of the Huge Primes material with some Kent schools, and at Aston University Engineering Academy in Birmingham, where we recently ran workshops with four groups of A-Level students on Pi Day 2023. |
| First Year Of Impact | 2018 |
| Sector | Digital/Communication/Information Technologies (including Software),Education,Culture, Heritage, Museums and Collections |
| Impact Types | Cultural,Societal |
| Description | ARA2 Applicable resurgent asymptotics: towards a universal theory |
| Amount | £330 (GBP) |
| Organisation | Isaac Newton Institute for Mathematical Sciences |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 09/2022 |
| End | 12/2022 |
| Description | COVID 19 Grant Extension Allocation University of Kent |
| Amount | £1,031,764 (GBP) |
| Funding ID | EP/V520718/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 06/2020 |
| End | 09/2021 |
| Description | Distinguished Researcher Visitor Scheme, School of Mathematics, University of New South Wales |
| Amount | $22,000 (AUD) |
| Organisation | University of New South Wales |
| Sector | Academic/University |
| Country | Australia |
| Start | 07/2018 |
| End | 01/2019 |
| Description | International Exchanges 2019 Cost Share JSPS: Generalized tau functions and cluster structures for birational difference equations |
| Amount | £12,000 (GBP) |
| Funding ID | IEC\R3\193024 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 03/2020 |
| End | 03/2023 |
| Description | Journal of Physics A sponsorship to support participants in conference at TSIMF, China |
| Amount | £800 (GBP) |
| Organisation | Institute of Physics Publishing |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 04/2019 |
| End | 04/2019 |
| Description | London Mathematical Society Undergraduate Research Bursary for Dr Philipp Lampe to supervise James Bradshaw in summer 2019 |
| Amount | £720 (GBP) |
| Funding ID | URB-18-19-19 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 07/2020 |
| End | 08/2020 |
| Description | London Mathematical Society Undergraduate Research Bursary for Prof Andrew Hone to supervise Tessa Wilks in summer 2017 |
| Amount | £720 (GBP) |
| Funding ID | 16-17 103 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 06/2017 |
| End | 08/2017 |
| Description | Public Engagement with Research |
| Amount | £1,000 (GBP) |
| Organisation | University of Kent |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 10/2017 |
| End | 10/2018 |
| Description | QR Uplift funding for project "Integrable lattices and soliton gases: from continuousto discrete, and from classical to quantum" |
| Amount | £23,000 (GBP) |
| Organisation | University of Kent |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 01/2022 |
| End | 07/2022 |
| Description | Research England Participatory Research fund for "Huge Primes" project with Institute for Research in Schools |
| Amount | £7,700 (GBP) |
| Organisation | University of Kent |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 02/2022 |
| End | 08/2022 |
| Description | Research in Pairs grant for visit of Dr Pavlos Kassotakis to Kent |
| Amount | £1,200 (GBP) |
| Funding ID | 42110 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 06/2022 |
| End | 08/2022 |
| Description | Science Faculty Impact Development Fund, University of Kent: outreach material for Institute for Research in Schools |
| Amount | £3,000 (GBP) |
| Organisation | University of Kent |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 05/2017 |
| End | 07/2017 |
| Description | Undergraduate Research Bursary for Jonathan Hobbs - "Huge primes: primality testing for Chebyshev-Lehmer primes" |
| Amount | £900 (GBP) |
| Funding ID | URB-2022-56 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 07/2022 |
| End | 08/2022 |
| Title | New sequences in the OEIS |
| Description | Sloane's Online Encyclopedia of Integer Sequences (OEIS) is the largest database of integer sequences and their properties. As a result of research by Hone and other members of the SeqFan mailing list, new results were obtained concerning sequence A269254, recording initial appearances of prime numbers in sequences of Chebyshev polynomials evaluated at integer arguments. At the request of Sloane, we wrote a paper in which we proved properties of infinite families of such sequences, published in J Integer Sequences and also available here https://arxiv.org/abs/1802.01793 (arXiv preprint). The paper resulted in updates and cross-referencing to about 20 OEIS sequences, including several new ones; full,details of all these sequences are listed in the paper, which includes hyperlinks. |
| Type Of Material | Database/Collection of data |
| Year Produced | 2018 |
| Provided To Others? | Yes |
| Impact | Funding is being sought for future research in which it is planned to use some of these sequences as a source of certified prime numbers, with potential applications in cryptography. |
| URL | https://oeis.org/A269254 |
| Description | CUMT |
| Organisation | China University of Mining and Technology (CUMT) |
| Country | China |
| Sector | Hospitals |
| PI Contribution | In April 2017 I gave two sets of lectures, entitled "Cluster algebras and integrable maps" and "Integrable and non-integrable equations with peakons" respectively, held jointly at CUMT and the Chinese Academy of Sciences at Beijing. We began preliminary discussions with Prof Liu and members of his group concerning integrable equations including anticommuting variables in a supersymmetry algebra. |
| Collaborator Contribution | I was hosted by Professor Qing Ping Liu and Dr Kai Tian. |
| Impact | No research outputs so far, but with Liu we were successfully awarded funding from Tsinghua International Mathematical Forum for a conference on Hamiltonian methods and algebraic structures for integrable systems, to be held at TSIMF, Sanya, China in April 2019. |
| Start Year | 2017 |
| Description | Chinese Academy of Sciences |
| Organisation | Chinese Academy of Sciences |
| Country | China |
| Sector | Public |
| PI Contribution | In April 2017 I gave two sets of lectures, entitled "Cluster algebras and integrable maps" and "Integrable and non-integrable equations with peakons" respectively, held jointly at CUMT and the Chinese Academy of Sciences in Beijing. We began preliminary discussions with members of the group of Prof Hu, concerning continued fractions and determinantal formulae for solutions of discrete integrable systems. |
| Collaborator Contribution | The lectures at the Chinese Academy were hosted by Prof Hu, and their was an in-kind financial contribution towards my travel and accommodation in Beijing. |
| Impact | No outputs yet, but I have maintained contact with Shi-Hao Li, a student of Hu who is going to take up a postdoctoral position at the University of Melbourne in 2018, and we are meeting at a conference in Sanya, China this April. |
| Start Year | 2017 |
| Description | DICATAM, Università degli Studi di Brescia |
| Organisation | University of Brescia |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | I have continued working with Dr Federico Zullo on coefficients of series expansions of Painleve transcendents/their tau functions. Zullo was formerly in Rome and worked with me as a postdoc in Kent with an INdAM-COFUND Marie Curie fellowship, but he now holds a permanent position in Brescia. Since visiting Brescia in November 2022, I have started to work with Zullo on a new analysis of Backlund transformations (BTs) for discrete integrable systems related to infinite Volterra and modified Volterra lattices, and associated Stieltjes continued fractions. |
| Collaborator Contribution | I was awarded funding (600 Euro) by Brescia to give some lectures there as a visiting professor in 2020/21, but this has had to be postponed due to the Covid-19 pandemic. Although I was worried it might be necessary to give the lectures online instead, when I was eventually able to travel again, I was able to visit Zullo in Brescia from 4-12 November 2022 and make use of this funding. |
| Impact | This is an ongoing collaboration at the interface of mathematics/theoretical physics.Existing outcomes were published and/or completed before Zullo moved to Brescia. |
| Start Year | 2020 |
| Description | La Trobe |
| Organisation | La Trobe University |
| Department | Department of Mathematics & Statistics |
| Country | Australia |
| Sector | Academic/University |
| PI Contribution | Work on discrete integrable systems; ordinary and partial difference equations and connections with cluster algebras. |
| Collaborator Contribution | Prof Reinout Quispel, Dr Peter van der Kamp, Dr Dinh Tran: discrete integrable systems, construction of reductions, Lax pairs, Poisson brackets and first integrals for ordinary and partial difference equations. Tran later became a postdoc at the University of Sydney in the group of Prof Nalini Joshi, so Hone visited her several times while on sabbatical in Sydney in 2017-2019. |
| Impact | https://arxiv.org/abs/1211.6958 https://arxiv.org/abs/1702.07047 |
| Start Year | 2010 |
| Description | Roma Tre |
| Organisation | Roma Tre University |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | Work on integrable Hamiltonian systems and their discretizations, and Painleve equations. |
| Collaborator Contribution | All the above topics, with Prof Orlando Ragnisco and Dr Federico Zullo. |
| Impact | https://arxiv.org/abs/1210.6822 (also many earlier publications with Ragnisco, Zullo et al.) |
| Description | Tsinghua |
| Organisation | Tsinghua University China |
| Country | China |
| Sector | Academic/University |
| PI Contribution | During my visit to Beijing in April 2017, I was invited to give a seminar at Tsinghua by Profs Youjin Zhang and Runliang Lin, and presented my results on continued fractions and transcendental numbers derived from nonlinear recurrence sequences. |
| Collaborator Contribution | I received an in-kind financial contribution to support the expenses for my visit. Lin also introduced me to some of his recent work on discrete bilinear equations, which are closely related to my work on maps obtained from cluster algebras. |
| Impact | No outputs yet, but I have been invited to make a return visit to Tsinghua in 2019. |
| Start Year | 2017 |
| Description | Universidad de la Rioja - Juan Luis Varona |
| Organisation | University of La Rioja |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | I was contacted by Juan Luis Varona by email in January 2017. He had extended some of my results on nonlinear recurrences that generate Engel series for transcendental numbers with explicit continued fraction expansions, by showing that the same methods were applicable to certain Pierce series, and published his work in J Number Theory. We agreed to collaborate, and after he found a further extension of Pierce series, I was able to generalize all of our previous results, as well as giving exact computation of the irrationality exponents of these numbers in many cases. We published our joint work in Monatshefte in 2018. |
| Collaborator Contribution | Juan Luis Varona had extended my earlier results on explicit continued fraction expansions to a different setting (Pierce series), and shared his results with me, suggesting that we could collaborate. After sharing some more of his preliminary results with me, we had a successful collaboration by email, and managed to publish our first paper together within two years. I hope to be able to visit Rioja to continue our collaboration in the near future. |
| Impact | A.N.W. Hone and J.L. Varona. Continued fractions and irrationality exponents for modified Engel and Pierce series. Monatshefte fur Mathematik, 16 pages, 2018. |
| Start Year | 2017 |
| Description | Universitat Politècnica de Catalunya |
| Organisation | Polytechnic University of Catalonia |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | Work on Somos sequences and algebraic curves; related integrable systems. |
| Collaborator Contribution | Prof Yuri Fedorov at UPC Barcelona, Spain: double covers and Prym varieties associated with Somos sequences and integrable systems in Hamiltonian mechanics. |
| Impact | See: https://arxiv.org/abs/1402.1102 https://arxiv.org/abs/1512.00056 (and details of papers on Researchfish). |
| Start Year | 2010 |
| Description | University of New South Wales |
| Organisation | University of New South Wales |
| Country | Australia |
| Sector | Academic/University |
| PI Contribution | From the middle of September 2017 I was a Visiting Professorial Fellow at the School of Mathematics and Statistics at UNSW, initially hosted for one year by Professors John Roberts and Igor Shparlinski, but my visiting position was extended to allow further stays up to the end of 2019. I have known Roberts and Shparlinski for several years and we have had informal discussions on a range of topics in the past. I have attended seminars and discussions in the Number Theory group as well as presenting two seminars myself, the first being "Curious Continued Fractions", concerning my recent results on transcendental numbers defined by nonlinear recurrence sequences, and the second on prime values of Chebyshev polynomials at integer arguments. I have also contributed to discussions at seminars in both the Statistics and the Applied Mathematics groups. New collaborative projects (with Roberts, Shparlinski and others) are underway. I also presented a 10-hour lecture course entitled "Discrete Integrable Systems and Cluster Algebras", in April-May 2018. |
| Collaborator Contribution | Initially, Roberts made a direct in-kind financial contribution to my project, by using some of his ARC grant money (AUD $10000) covering almost three months' rent from October 2017, during my initial stay in Sydney; the following year he provided more supporting funds (over $5000) from his grant. He also provided his own expertise on the topic of discrete dynamics over finite fields, through our developing collaboration. As part of collaborative discussions and seminars on the same topic, and concerning linear recurrence sequences over residue rings, I have also had the benefit of the expertise of Shparlinksi and Ostafe, and that of Harvey on computational number theory; I have been doing extensive numerical computations of birational maps iterated over finite fields, and it has been invaluable to have expert advice on this. At the beginning of September 2018 I was awarded additional funding from the School of Mathematics & Statistics under the Distinguished Researcher Visitor Scheme, to support me over an additional three months at UNSW. The direct in-kind contribution of the Mathematics & Statistics department amounted to a total commitment of AUD $22000 under this scheme (including tax and on-costs). |
| Impact | So far in 2018 I wrote a paper on the appearance of primes in certain linear recurrence sequences with two other collaborators (Jeffery and Selcoe), "On a family of sequences related to Chebyshev polynomials", preprint arXiv:1802.01793v2 (appeared in J. Integer Sequences, 2018). In preparing the paper I have received many helpful comments from Harvey, Roberts and Shparlinski at UNSW. I have also published a paper with Varona (Rioja) on continued fractions for Engel and Pierce series (appeared in Monatshefte fur Mathematik, 2018) based on an extension of results which I presented in the Number Theory seminar at UNSW. |
| Start Year | 2017 |
| Description | University of Tokyo - Generalized tau functions and cluster structures for birational difference equations |
| Organisation | University of Tokyo |
| Country | Japan |
| Sector | Academic/University |
| PI Contribution | We have been awarded funding of £12000 from Royal Society under the International Exchanges Cost Share programme with JSPS in Japan, which will fund visits by members of Hone's team at Kent, as well as his colleagues Dr Clelia Pech and Prof Jing Ping Wang, to make visits to Graduate School of Mathematical Sciences at the University of Tokyo. There we are collaborating with a group headed by Prof Ralph Willox, using cluster algebras to describe the structure of birational maps. It was impossible to use the funding during the original period of the grant, but the Royal Society gave us a 12-month No Cost Extension until 17 March 2023, allowing a 2-week visit by Hone with PhD student Joe Harrow in January 2023. We also hosted Prof Ralph Willox and Dr Takafumi Mase in Canterbury: they made a visit to Kent from 8-22 February 2023, and on 13th February gave talks in our Mathematics colloquium, entitled "The Laurent property of the Burchnall-Chaundy polynomials" and "The Laurent property, irreducibility and coprimeness of non-integrable partial difference equations", respectively. |
| Collaborator Contribution | The group in Tokyo has a great deal of expertise in integrable systems and difference equations. Prof Ralph Willox and Dr Takafumi Mase started by organizing online seminars given by Hone during the covid pandemic, and subsequently hosted Hone and PhD student Joe Harrow at the University of Tokyo from 3-16 January 2023, and from initial discussions that continued in their follow-up visit to Kent in February, we have begun working on two different projects: the first concerns discrete analogues of symmetric Painleve equations (including a discrete PIV equation) and polynomial tau functions arising by reduction of the discrete Hirota equation; and the second is about Laurentification of general QRT maps and other related birational maps; |
| Impact | See under additional funding. The collaboration has only just begun but joint papers are planned. |
| Start Year | 2020 |
| Description | "Huge Primes" Masterclass by Andy Hone at Barton Court Grammar School, Canterbury, 18th November 2022. |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Schools |
| Results and Impact | Masterclass on RSA Cryptography and the applications of integer sequences and number theory (especially, large prime numbers) to a group of 60-70 pupils taking A-Level Mathematics in Years 12 & 13. The students engaged very enthusiastically in the challenges they were given about encrypting messages. This event was a pilot of some of the "Huge Primes" material being developed with the Institute for Research in Schools, which we are planning to roll out as an online project to all UK schools over the next year. The Head of Mathematics at Barton Court Grammar School requested future maths-related enrichment activities for A-Level students to take place at the University of Kent in 2023. |
| Year(s) Of Engagement Activity | 2022 |
| Description | ARA Applicable resurgent asymptotics: towards a universal theory, programme at Isaac Newton Institute, Cambridge |
| 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 | The ARA programme at INI, Cambridge was split between online activities between March-June 2021 (due to the pandemic), and the second part in September-December 2022, involving live participation. In the first part Prof Hone led an online discussion on Painleve equations (25 May 2021) and chaired a talk by Its, then spent three weeks in Cambridge in September (giving the talk "Variational principles in field theory" in the Moller Institute, 20th September, and the ARA seminar "Continued fractions, orthogonal polynomials and hyperelliptic curves" on 23rd September), returning again 20th-21st October for part of a workshop on Dispersive Waves. |
| Year(s) Of Engagement Activity | 2021,2022 |
| URL | https://www.newton.ac.uk/event/ara2/ |
| Description | Contribution to 16th Int'l Conf on Foundations of Computer Science (FCS'20), part of CSCE'20 - The 2020 World Congress in Computer Science, Computer Engineering, & Applied Computing, July 27-30, 2020 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I contributed the paper "ECM Factorization with QRT Maps" on my research results using dynamical systems for factorization of large integers, to a large computer science congress, in order to make my results known to academic and professional practitioners outside my discipline. It has led to me getting in touch with computer scientists working in cyber security, including a headhunter for Google, and I have plans to develop new algorithms based on the results and related funding applications. |
| Year(s) Of Engagement Activity | 2020 |
| URL | https://www.americancse.org |
| Description | Contribution to ISSAC 2020, International Symposium on Symbolic and Algebraic Computation, 20-23 July 2020 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The paper "Efficient ECM factorization in parallel with the Lyness map" was contributed to the ISSAC conference on symbolic computation (online due to the coronavirus pandemic), and published in the proceedings by Association for Computing Machinery. The purpose was to present some of my results on discrete dynamics in finite fields/residue rings to practitioners outside my area, with a view to finding the most appropriate potential applications of these theoretical techniques. I got some useful feedback from the reviewers. |
| Year(s) Of Engagement Activity | 2020 |
| URL | https://www.issac-conference.org/2020/ |
| Description | I'm a Scientist Get me Out of Here - Hydrogen Zone activity in August/September 2021 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Schools |
| Results and Impact | I'm a Scientist Get me Out of Here was a series of online chats for school pupils. I participated as one of 23 scientists in the Hydrogen Zone (of which the main focus was physical sciences), who engaged in a series of online chat sessions with a total of 194 school pupils during August-September 2021. We were asked a rapid fire sequence of questions about our experiences, daily work and interests as scientists, and had to reply to these and other general science questions posed by the participants at a range of UK schools. |
| Year(s) Of Engagement Activity | 2021 |
| URL | http://hydrogen21.imascientist.org.uk |
| Description | IRIS Pilot Project Launch Workshop |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Schools |
| Results and Impact | Afternoon workshop attended by pupils and teachers at 5 Kent schools held at Simon Langton Boys' Grammar School, Canterbury. The topic of discrete dynamics was introduced to the group via a worksheet with a set of interactive exercises. Once they understood the topic, the contribution that pupils can make to research in the project was described to all the participants, with a plan of how to roll it out across the UK after the initial trial year 2017-18. Money from University of Kent's Science Faculty Impact Fund (£3000) is being provided to the Institute for Research in Schools (IRIS) to develop the materials from the workshop into an online resource for schools. The materials will be freely available on the IRIS website so that pupils across the UK can take part in the project. The workshop was organized by Professor Becky Parker (IRIS Director, Head of Physics at Simon Langton Boys Grammar and Visiting Professor in Physics at QMUL). I made links with several teachers and pupils from the different schools involved, who were interested in taking part in the project. |
| Year(s) Of Engagement Activity | 2017 |
| URL | http://www.researchinschools.org/ |
| Description | Invited talk in special session on Mathematical Physics, BMC/BAMC, Glasgow (online), 6-9 April 2021 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | The joint British Mathematical Colloquium/British Applied Mathematical Colloquium (BMC/BAMC) is the largest annual UK-based meeting covering all areas of Mathematics, but also has several plenary international speakers. This was postponed from the originally planned live event in 2020, and was moved to a highly effective online platform in 2021. I gave the invited talk "Heron triangles with two rational medians and Somos-5 sequences" in the Mathematical Physics special session: it was the first time I presented these new results to a wide audience (before they were published), and the questions I was asked were very helpful as they encouraged me to discover new exact formulae before I wrote up the final version of the paper. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://sites.google.com/view/bmcbamc2021/home |
| Description | Lectures in Beijing, April 2017 |
| 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 | In April 2017 I gave two sets of lectures, entitled "Cluster algebras and integrable maps" and "Integrable and non-integrable equations with peakons" respectively, held jointly at CUMT and the Chinese Academy of Sciences in Beijing. The lectures were attended by staff and postgraduate students from both institutions, as well as a number of mathematicians from other parts of China who travelled to Beijing specifically to hear my lectures and those of my collaborator Dr Vladimir Novikov (Loughborough), who spoke about the symmetry approach to classification of integrable equations. I also gave a seminar at Tsinghua University, where I was invited by Profs Youjin Zhang and Runliang Lin; there I presented my latest results about discrete dynamical systems that generate transcendental numbers with an explicit continued fraction expansion. During the visit I discussed future collaborative projects with Lin and Prof Qing Ping Liu, my host at CUMT. |
| Year(s) Of Engagement Activity | 2017 |
| Description | Lectures on cluster algebras and discrete integrability by Hone at UNSW, Sydney, April/May 2018 |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | Between 27th April - 11th May I gave a course of five two-hours lectures introducing the theory of cluster algebras and related discrete integrable systems, culminating on my latest results on the algebraic entropy classification of cluster maps. The course was attended by postgraduate students and staff in Mathematics at UNSW and University of Sydney, as well as some international visitors. This led to further discussions with Ostafe, Roberts and Shparlinski concerning Markoff numbers and related dynamical systems over finite fields, with a view to collaborating on using cluster theory to obtain new results in the future. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://www.maths.unsw.edu.au/seminars/2018-04/cluster-algebras-and-integrable-maps-lecture-1 |
| Description | Local TV interview - KMTV - with Hone about 80th anniversary of breaking of Enigma code, 9th July 2021 |
| Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Public/other audiences |
| Results and Impact | Hone was interviewed as a short item on local Kent television news (KMTV) about the the 80th anniversary of the Enigma Code being cracked by Alan Turing and his team at Bletchley Park. (The short clip is available to be seen at time 18:10 on the web recording linked in the url below.) |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://www.kentonline.co.uk/kmtv/video/kent-tonight-friday-9th-july-2021-34888/ |
| Description | Loughborough Geometry & Mathematical Physics Seminar |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Talk by Hone entitled "Continued fractions and Hankel determinants from hyperelliptic curves" on 27th November 2019 in Loughborough. I also examined a PhD thesis and attended the Integrable Day workshop on 29th. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www.lboro.ac.uk/departments/maths/news-events/geometry-mathematical-physics/ |
| Description | Mathematics PhD seminar talk by Joe Pallister in Leicester, December 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Postgraduate students |
| Results and Impact | On 4th December, PhD student Joe Pallister from Kent was visiting Leicester, where he gave a seminar entitled "Friezes and cluster algebras" to postgraduates in Mathematics, in which he presented an introduction to frieze patterns and the related theory of cluster algebras. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www2.le.ac.uk/departments/mathematics/extranet/for-current-students/PG-students/phd-seminars... |
| Description | Maths talk at Glenaeon School, Sydney, Australia in July 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Schools |
| Results and Impact | On 2nd July 2019 I gave a talk to Year 12 students specializing in Advanced Mathematics for HSC (the Australian equivalent of A-Levels). I explained various aspects of my mathematical research to them, and described where a university degree in Mathematics can lead. Some of the students said they were enthusiastic about doing further mathematical study. |
| Year(s) Of Engagement Activity | 2019 |
| Description | Maths talk at Glenaeon School, Sydney, Australia in September 2018 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Schools |
| Results and Impact | Talk for Year 12 Advanced Mathematics students at Glenaeon School, Sydney, Thursday 20th September 2018. I discussed discrete and continuous dynamical systems, and their use for problems in physics, biology and number theory. The talk was interactive, with answers and results of calculations solicited from the audience. I used examples from my own research, in particular my current EPSRC fellowship project. It was well received, and one student told me later that he now wanted to do a mathematical degree at university after hearing my talk. |
| Year(s) Of Engagement Activity | 2018 |
| Description | NSPCC Number Day at Hampton Primary, Herne Bay in February 2020 |
| 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 | The NSPCC invited me to lead an assembly for 90 pupils in Year 5 at Hampton Primary in Herne Bay, Kent, as part of the fundraising activities for their national Number Day, on 7th February 2020. Together with my colleague Dr Steffen Krusch and Undergraduate Ambassador Georgia Ashley, we ran a workshop entitled "Maths in Nature" which covered Fibonacci's rabbit breeding puzzle, golden mean spirals in shells and pine cones, and the growth of flowers and plants. The children were very enthusiastic, and the NSPCC press officers took photos for their publicity and local media. Funds raised from Number Day will help support vital NSPCC services such as Speak Out Stay Safe, a program which sees the NSPCC visit primary schools across Kent to teach children about the signs of abuse and neglect and who to talk to if they are worried. Hannah Gregory, a teacher at the school said: "Hampton Primary School have had a fantastic Number Day today to help raise funds for the NSPCC. All day the children have been participating in a variety of fun maths activities ranging from weighing and making delicious recipes to tackling times table challenges. Year five pupils also thoroughly enjoyed Andy Hone's visit and taking part in the engaging maths activities that he arranged." |
| Year(s) Of Engagement Activity | 2020 |
| URL | https://www.nspcc.org.uk/what-you-can-do/charity-runs-cycles-and-challenges/social-and-special-event... |
| Description | Number Theory Seminar UNSW by Hone, October 2017 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Seminar "Curious continued fractions" by Hone in the UNSW Number Theory Seminar. This was my first presentation to the group of Prof Igor Shparlinski. |
| Year(s) Of Engagement Activity | 2017 |
| URL | https://www.maths.unsw.edu.au/seminars/2017-10/curious-continued-fractions |
| Description | Number theory seminar at UNSW, June 2019 by Andrew Hone |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Talk by Hone at UNSW on 19th June 2019: "Hankel determinants and continued fractions for hyperelliptic functions" in the Number Theory seminar. Afterwards with Prof John Roberts I discussed analysis of some of the maps considered over finite fields, which could be incorporated to our work in progress in which we have developed ways to identify algebraic integrability in finite fields. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www.maths.unsw.edu.au/seminars/2019-06/hankel-determinants-and-continued-fractions-hyperelli... |
| Description | Physics seminar in Lincoln by Dr Theodoros Kouloukas |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Physics seminar by Kouloukas in the School of Mathematics & Physics, University of Lincoln, 19th February 2020. Title: "Integrability properties of generalized Lotka-Volterra systems". The talk concerned his latest results on integrable Hamiltonian systems obtained in collaboratíon with Hone and Christodoulidi (Lincoln), published in J Computational Dynamics. It was an opportunity to discuss further work with one of his collaborators. |
| Year(s) Of Engagement Activity | 2020 |
| Description | Pi Day at AEUA, Birmingham: "Huge Primes" Masterclasses on 14th March 2023 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Schools |
| Results and Impact | With support from the Institute for Research in Schools (IRIS), Hone was invited to trial material for the "Huge Primes" project at Aston University Engineering Academy (AUEA), as part of celebrations for Pi Day 2023. Four Mathematics Masterclasses on RSA Cryptography were run with different groups of Year 12 students from AUEA and Aston Manor (another Birmingham school nearby), to a total of around 80 students. The students were very enthusiastic about the sessions and worked really hard on the encryption and code-breaking challenges they were offered. Afterwards, Hone met with Ayan Farah, Head of Mathematics at AUEA, and Katy Glazer (IRIS) to plan future developments for running a pilot version of the Huge Primes project at AUEA, before rolling it out across all UK schools a year later. |
| Year(s) Of Engagement Activity | 2023 |
| Description | Representation Theory Seminar by Dr Philipp Lampe in Bielefeld, Germany, June 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | On 7th June 2019, Lampe presented the talk "The growth of real seeds in cluster theory and a group theoretic determinant" to a group of international experts and postgraduate students in the Representation Theory Seminar at the University of Bielefeld, Germany. He held discussions with a view to future collaboration with members of the group. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www.math.uni-bielefeld.de/birep/seminar.php |
| Description | Research visit and seminar at DICATAM, Sezione di Matematica Università degli Studi di Brescia, 4-12 November 2022. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Hosted by Dr Federico Zullo, with whom I carried out new research on Backlund transformations and Lax pairs associated with Stieltjes continued fractions for hyperelliptic solutions of infinite Volterra and modified Volterra lattices. DICATAM Mathematics Seminar, 9th November: "Frazioni Continue, Curve Algebriche, Sistemi Dinamici" (in Italian). |
| Year(s) Of Engagement Activity | 2022 |
| Description | Royal Institution Mathematics Masterclass in Kent, January 2020 |
| 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 a two-hour masterclass at the University of Kent on Saturday 11th January for the most mathematically precocious 6th form students selected by secondary schools in Kent. The title was "Elliptic Curves - Marvels and Mysteries". I gave the students challenging problems in number theory, concerning the solution of Diophantine equations, and gave them an outline of the connection between elliptic curves and Wiles' proof of Fermat's last theorem, as well as the conjecture of Birch and Swinnerton-Dyer, one of the million-dollar Clay Millenium Prize problems. |
| Year(s) Of Engagement Activity | 2020 |
| URL | https://www.rigb.org/education/masterclasses/our-networks/secondary-network |
| Description | Royal Institution Mathematics Masterclasses (Kent) |
| 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 | Masterclass at University of Kent, 4th February 2017, covering discrete dynamics and integer sequences, with applications in physics, biology and number theory. Also, on February 25th 2023, March 11th 2023 (and other Saturdays in Spring Term), Hone and colleagues from SMSAS, Kent ran online Mathematics Masterclasses for pupils from schools in the Kent and London area, supported by the Royal Institution of Great Britain. Specifically, Hone was running chat sessions and breakout rooms involving 160-200 Year 12 students, to support the presentations given by colleagues Dr Clare Dunning (on combinatorics and square ice models in mathematical physics) and Dr Bas Lemmens (on the Google PageRank algorithm). These Masterclasses were originally set up by Hone at the University of Kent as smaller classroom events, about 15 years ago, but have grown substantially. During the recent pandemic, they moved online, and have become even more successful at reaching a large number of students. |
| Year(s) Of Engagement Activity | 2017,2023 |
| Description | School visit (Ashford) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Schools |
| Results and Impact | Norton Knatchbull Grammar School, Ashford, 20th January 2017. Motivational talk on studying mathematics at university, illustrated from my past and current research projects. The talk addressed a large group of pupils studying mathematics in years 11-13 (GCSE & A-Level). |
| Year(s) Of Engagement Activity | 2017 |
| Description | Seed Funding meeting at the French Embassy, London, 27-28 October 2022 |
| 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 | With Prof Pol Vanhaecke (Poitiers), in 2021 Hone successfully applied for Ambassade de France au Royaume Uni/British Council Seed Funding to hold France-UK collaborative meeting with a group of approximately 10 mathematicians from the two countries, which was held at the French Embassy in London, 27-28 October 2022. The discussions and presentations concerned current and future funding proposals on nonlinear waves and integrable systems, including a COST European network proposal that Hone had submitted shortly before, and follow-up projects leading out of Hone's EPSRC fellowship. Vanhaecke was one of the co-proposers of the COST Action "Coherent Structures, Lattice Equations and Soliton Solutions" (COSTLESS), along with Prof Tamara Grava (Bristol/SISSA). In addition to discussing the latter, we looked at ways to extend to wider research networks in this area and foster closer collaboration and coordinated funding applications, and the French Embassy research staff explained ways in which they can support specific France-UK joint ventures. |
| Year(s) Of Engagement Activity | 2022 |
| Description | Seminar (Zoom webinar) at Graduate School for Mathematical Sciences, University of Tokyo, 22nd July 2020 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Zoom seminar "Efficient ECM factorization in parallel with the Lyness map" and discussion with researchers and PhD students in the group of Professor Ralph Willox in Tokyo. We discussed how to develop our Royal Society-funded project on tau functions and generalized cluster structures, which is an offshoot of my EPSRC fellowship.Also I was put in touch with Kakei at Rikkyo University who is interested in similar problems related to discrete dynamics overfinite fields/residue rings. |
| Year(s) Of Engagement Activity | 2020 |
| Description | Seminar by Hone at Graduate School of Mathematical Sciences, University of Tokyo, 28 October 2021 (online). |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Invited talk: "Deformations of cluster mutations and invariant presymplectic forms" - this described my recent work with Theodoros Kouloukas, relating to tau functions, cluster variables and the Laurent property, which is the subject of the Royal Society International Exchange project between my group and the group of Prof Ralph Willox in Tokyo. |
| Year(s) Of Engagement Activity | 2021 |
| Description | Seminar by Hone in Selected Topics in Mathematics - Online Edition, Liverpool, 8th February 2023 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This online seminar series, organized by Dr Matty van Son and Dr Oleg Karpenkov (Liverpool), concerns topics related to continued fractions, approximation theory and geometry of numbers. It attracts an international audience, some of whom asked questions and made contacted afterwards with requests for more information on the talk "Continued fractions and hyperelliptic curves". |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://pcwww.liv.ac.uk/~karpenk/seminar/seminar.html |
| Description | Seminars in Melbourne, May 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I visited Dr Shi-Hao Li and the group of Prof Peter Forrester in the School of Mathematics & Statistics, University of Melbourne for a few days, where I gave a seminar "Chebyshev polynomials, Lehmer numbers and huge primes" on 15th May 2019, and attended an outreach meeting. Li was particularly interested in potential future collaboration on higher dimensional analogues of continued fractions and orthogonal polynomials. Afterwards I spent a few days visiting my collaborator Prof Reinout Quispel at La Trobe University, where I gave another version of the same seminar. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://ms.unimelb.edu.au/events/details?event=12666 |
| Description | Talk by Hone at "Celebrating Redundancy" in honour of Prof Reinout Quispel, La Trobe University, Melbourne, 21st October 2022 (online) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Online talk at hybrid workshop in Melbourne to celebrate the retirement of Prof Reinout Quispel. Talk by Hone: "Continued fractions and hyperelliptic curves". I had online discussions about my own and the other talks with some of the participants, and one of them contacted me afterwards about doing some journal refereeing on a related topic. |
| Year(s) Of Engagement Activity | 2022 |
| Description | Talk by Hone at LMS Undergraduate Summer School, Heriot-Watt University, 24th August 2022 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Undergraduate students |
| Results and Impact | Hone made a trip to Edinburgh from 23-27 August 2022, visiting colleagues in the Mathematics Department at the University of Edinburgh and staying on the Heriot-Watt University campus as an invited speaker at the London Mathematical Society (LMS) Undergraduate Summer School. He gave a talk about his research in integrable systems and nonlinear waves, entitled "Peakons - a new paradigm in soliton theory", to an audience of approximately 50 specially selected Mathematics undergraduates from UK universities. This was followed by an outing with the students to the Union Canal, to visit the Scott Russell viaduct where one of these special waves (a soliton) was first observed in 1834 by the Scottish naval architect John Scott Russell. It was great fun to interact with the students and answer their questions about many different aspects of mathematics. |
| Year(s) Of Engagement Activity | 2022 |
| URL | http://www.lms.macs.hw.ac.uk/ |
| Description | Talks at Sophia Mundi School, Melbourne, Australia in May 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Schools |
| Results and Impact | On 17th May 2019, I gave talks to two groups of students in Year 11 & 12 at Sophia Mundi School in Melbourne. In the morning I led the Theory of Knowledge class, where a group of approximately 20 students were starting a new topic, namely the nature of mathematical knowledge. They asked me many questions, and some of the students were surprised to find that their view of mathematics changed in the course of the discussion, particularly those who were not specializing in higher mathematics or science. In the afternoon I took a group of about 10 students, consisting of the students pursuing further mathematical studies, and we discussed university courses and project work in mathematics. |
| Year(s) Of Engagement Activity | 2019 |
| Description | Talks in Beijing and Sanya, China in April 2019 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | In the first week of April 2019 I was hosted at CUMT, Beijing where I gave three hours of lectures on integrable systems and singularity analysis to postgraduate students and staff, and gave a seminar on primes in Chebyshev polynomial sequences at Tsinghua University. The following week I was one of the organizers of the conference on Hamiltonian Methods and Algebraic Structures for Integrable Systems (8th-12th April) where around 50 participants were hosted at TSIMF, Sanya with financial support from the Yau Mathematical Sciences Center. I led a plenary session on open problems in Hamiltonian integrable systems. This led to plans to tackle further research problems on algebro-geometric methods in mechanics with Profs Robert Conte and Yuri Fedorov. Also at Tsinghua I met Prof Zhedanov who made me aware of connections of my recent results with representation theory of double affine Hecke algebras. |
| Year(s) Of Engagement Activity | 2019 |
| URL | http://ymsc.tsinghua.edu.cn/en/content/lists/180.html# |
| Description | Two Royal Institution Mathematics Masterclasses (Spring term 2019, Kent) |
| 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 | Two Mathematics Masterclasses at the University of Kent, Saturday 9th March, reaching approximately 115 students from schools in Kent. 11.00-13.15, Year 9 workshop (approx 70 students), Dynamics of Population Growth: explored discrete dynamics and its appearance in biology, physics and number theory, including a brief introduction to chaos in the logistic map. 14.00-16.30, Year 12/13 workshop (approx 45 students), Elliptic Curves - Marvels and Mysteries: introduced algebraic curves viewed as Diophantine equations, the group structure on elliptic curves and how this appears in Wiles' proof of Fermat's Last Theorem, and the conjecture of Birch & Swinnerton-Dyer (one of the Clay Millenium Prize problems). |
| Year(s) Of Engagement Activity | 2019 |
| Description | Two Royal Institution Mathematics Masterclasses in Kent, March 2019, given by RA Dr Philipp Lampe |
| 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 | On Saturday 2nd March, Lampe gave two Mathematics Masterclasses at the University of Kent. In the morning he covered Continued fractions and the Pell equation with approximately 50 Year 9 pupils from schools across Kent, and in the afternoon roughly the same number of Year 12/13 students came to take part in his workshop on Pólya's enumeration theorem. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://blogs.kent.ac.uk/smsasoutreach/ |
| Description | Two talks by Kouloukas at Edinburgh Mathematical Physics Seminar, January 2018 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | On 31st January 2018 Kouloukas gave two talks at the joint University of Edinburgh/Heriot-Watt Mathematical Physics Seminar meeting: a pre-talk entitled "Introduction to discrete integrable systems" which introduced his subject matter to PhD students, and a specialized seminar aimed at the general audience on "Yang-Baxter maps and relativistic collisions", in which he presented his latest research results on this topic. During his visit he also discussed potential future collaborative projects with Prof Anastasia Doikou and Dr Panagiota-Maria Adamopoulou. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://empg.maths.ed.ac.uk/HTML/AY2017Seminars.html |
| Description | Two-week research exchange visit and seminars at University of Tokyo by Prof Andrew Hone & PhD student Mr Joe Harrow, January 2023 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Research visit by Hone & Harrow to group of Prof Ralph Willox at University of Tokyo as part of Royal Society International Exchange project. Seminar talks on 11th January: "Determinantal expressions for Ohyama polynomials" (Harrow), "Discrete dynamics, continued fractions and hyperelliptic curves" (Hone); and 13th January: "An infinite sequence of Heron triangles with two rational medians" (Hone) - these attracted not only staff and student members of Willox's group, but also other integrable systems experts from elsewhere (Kakei, Matsuno, Ohta, Sakai). This resulted in a follow-up visit by Willox and Mase to Kent in February 2023, when we developed our collaboration further. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.ms.u-tokyo.ac.jp/seminar/2023/sem23-009_e.html |
| Description | UNSW Number Theory Seminar by Hone, March 2018 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Talk "On a family of sequences related to Chebyshev polynomials" by Hone, UNSW Number Theory Seminar, 28th March 2018. I had the opportunity to discuss my results with Prof Igor Shparlinski, Dr Alina Ostafe, and Dr David Harvey, who made several useful comments regarding different theoretical and computational approaches to the problems considered in the talk. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://www.maths.unsw.edu.au/seminars/2018-03/family-sequences-related-chebyshev-polynomials |
| Description | Visit to Mathematical Sciences Institute, ANU Canberra, Australia in March 2019 by Prof Andrew Hone |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I was hosted by Prof Peter Peter Bouwknegt, the director of the MSI, a department of the ANU Canberra, 26-28 May 2019. The invitation was arranged by Dr Geoffrey Campbell, with whom I have been in contact via the Number Theory group that he runs on Linkedin. I had the opportunity to talk to Prof Murray Batchelor and meet Prof Rodney Baxter, an eminent mathematical physicist, and gave a Physics seminar in which I presented my work on cluster algebras and discrete.dynamics, and a Mathematics colloquium in which I discussed my recent results on appearance of primes in sequences of values of Chebyshev polynomials of the 3rd and 4th kinds. |
| Year(s) Of Engagement Activity | 2019 |
| Description | What's Your Angle, Science Museum |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Public/other audiences |
| Results and Impact | 1400 people came to What's Your Angle, the London Mathematical Society 150th Anniversary Festival at the Science Museum in London, 24th-29th November 2015. I took part in Kent's contribution, which was an interactive presentation of solitons and nonlinear waves. The material was particularly well received by the groups from schools that I presented it to. |
| Year(s) Of Engagement Activity | 2015 |
| URL | https://www.lms.ac.uk/2015/footfall-festival |
| Description | Yaroslavl online seminar on integrable systems and related topics, 22nd April 2020 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Two-hour research seminar hosted via Zoom in Yaroslavl Centre of Integrable Systems, Russia, entitled "Efficient ECM factorization in parallel with the Lyness map", which reached an international audience of researchers and postgraduate students (about 30 total). There were some interesting questions afterwards and I plan to develop the research in the future with colleagues interested in using dynamical systems to find new cryptographic algorithms. |
| Year(s) Of Engagement Activity | 2020 |
| URL | https://www.youtube.com/watch?v=PM0NbnINyTQ |