The theory of (ordinary and partial) differential equations is well-established and to some extent standardised. By contrast, the theory of difference equations, while more fundamental, has until recently been in its infancy, in spite of a major effort at the beginning of the 20th Century by N"orlund and the school of G.D. Birkhoff to establish the linear theory. Discrete systems can appear in two main guises: in the first case the independent variable is discrete, taking values on a lattice (e.g. finite-difference equations, such as recurrence relations and dynamical mappings), in the second case the independent variable is continuous (e.g. analytic difference equations and even functional equations).
Very recently, however, mainly through advances in the theory of exactly integrable discrete systems and the theory of (linear and nonlinear) special functions, the study of difference equations has undergone a true revolution. For the first time good and interesting examples of nonlinear difference equations admitting exact, albeit highly nontrivial, solutions were found and this has led to the formulation of novel approaches to the classification and treatment of such equations. Thus, an area has developed where several branches of mathematics and physics, that are usually distinct, come together: complex analysis, algebraic geometry, representation theory, Galois theory, spectral analysis and the theory of special functions, graph theory, and difference geometry.
Looking at recent developments in these various fields one observes that many of the areas that are clearly on the verge of major breakthroughs have an intimate connection with discrete integrable systems. Such fields comprise: the algebraic geometry of rational surfaces and birational maps, difference Galois theory and isomonodromic deformation theory, Diophantine problems in number theory and p-adic analysis, quantum and non-commutative discrete integrable systems, representation theory of quantum algebras and associated cluster algebras as well as special functions, difference geometry and symmetries and conservation laws of discrete systems. Bringing together experts from these fields for a substantial period of time the programme aims at bringing about important cross-fertilisation of ideas and methodologies, and thereby substantial new advances in the theory of difference equations.
Click here to download the programme's final scientific report
Title | Year | Programme |
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The $\widehat{\mathfrak{sl}}(n)_{k}$-WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomologyAuthors: Catharina Stroppel, Christian Korff |
2009 | DIS |
Analytic solutions and integrability for bilinear recurrences of order sixAuthors: ANW Hone |
2008 | DIS |
Direct "delay" reductions of the Toda hierarchyAuthors: Nalini Joshi, Paul Spicer |
2008 | DIS |
Semi-classical Laguerre polynomials and a third order discrete integrable equationAuthors: Frank Nijhoff, Paul Spicer |
2008 | DIS |
Integrable non-QRT mappings of the planeAuthors: Nalini Joshi, Pavlos Kassotakis |
2008 | DIS |
Elliptic solutions of ABS lattice equationsAuthors: James Atkinson, Frank Nijhoff |
2008 | DIS |
A constructive approach to the soliton solutions of integrable quadrilateral lattice equationsAuthors: Frank Nijhoff, James Atkinson |
2008 | DIS |
Growth of degrees of integrable mappingsAuthors: Peter van der Kamp |
2008 | DIS |
Okamoto's space for the first Painlevé equation in Boutroux coordinatesAuthors: Nalini Joshi, Hans Duistermaat |
2008 | DIS |
On the non-multiaffine consistent around the cube lattice equationsAuthors: M Nieszporski, Pavlos Kassotakis |
2008 | DIS |
On a nonlinear integrable difference equation on the square 3D-inconsistentAuthors: Ravil Yamilov, Decio Levi |
2008 | DIS |
The generalised symmetry method for discrete equationsAuthors: Decio Levi, Ravil Yamilov |
2008 | DIS |
Soliton solutions for ABS lattice equations: I Cauchy matrix approachAuthors: Jarmo Hietarinta, Frank Nijhoff, James Atkinson |
2008 | DIS |
Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clustersAuthors: Simon Ruijsenaars, Jonathan Nimmo |
2008 | DIS |
Symmetries and integrability of discrete equations defined on a black-white latticeAuthors: Pavlos Xenitidis, Vassilios Papageorgiou |
2008 | DIS |
Singular finite-gap operators and indefinite metric. I.Authors: P Grinevich, Sergey Novikov |
2008 | DIS |
Lagrangian multiforms and multidimensional consistencyAuthors: Frank Nijhoff, Sarah Lobb |
2008 | DIS |
Integrability, analyticity, isochrony, equilibria, small oscillations and Diophantine relationsAuthors: Francesco Calogero |
2008 | DIS |
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebrasAuthors: Matteo Petrera, ANW Hone |
2008 | DIS |
Linear quadrilateral lattice equations and multidimensional consistencyAuthors: James Atkinson |
2008 | DIS |
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa$-$Holm type equationAuthors: J Szmigielski, ANW Hone, H Lundmark |
2008 | DIS |
On the Neimark$-$Sacker bifurcation in a discrete predator-prey systemAuthors: ANW Hone, MV Irle, GW Thurura |
2008 | DIS |
Note on the Poisson structure of the damped oscillatorAuthors: M Senthilvelan, ANW Hone |
2008 | DIS |
On the non-integrability of the Popowicz peakon systemAuthors: MV Irle, ANW Hone |
2008 | DIS |
On non-standard numerical integration methods for biological oscillatorsAuthors: ANW Hone |
2008 | DIS |
A self-adaptive mesh method for the Camassa$-$Holm equationAuthors: Yasuhiro Ohta, K Maruno, BF Feng |
2008 | DIS |
Holomorphic curves with shift-invariant hyperplane preimagesAuthors: R Korhonen, K Tohge, Rod Halburd |
2008 | DIS |
Darboux transformations for linear operators on two dimensional regular latticesAuthors: M Nieszporski, Adam Doliwa |
2008 | DIS |
Singular finite-gap operators and indefinite metric II$^3$Authors: Sergey Novikov, P Grinevich |
2008 | DIS |
Desargues maps and the Hirota$-$Miwa equationAuthors: Adam Doliwa |
2008 | DIS |
Asymmetric integrable quad-graph equationsAuthors: Peter Hydon, Claude Michel Viallet |
2008 | DIS |
Coisotropic deformations of algebraic varieties and integrable systemsAuthors: Boris Konopelchenko, G Ortenzi |
2008 | DIS |
Yang$-$Baxter maps associated to elliptic curvesAuthors: AG Tongas, Vassilios Papageorgiou |
2008 | DIS |
Multisoliton solutions to the lattice Boussinesq equationAuthors: Cheng Zhang, Jarmo Hietarinta |
2008 | DIS |
Difference Picard theorem for meromorphic functions of several variablesAuthors: R Korhonen |
2008 | DIS |
Non-decreasing functions, exceptional sets and generalised Borel lemmasAuthors: RJ Korhonen, Rod Halburd |
2008 | DIS |
Lagrangian multiform structure for the lattice KP systemAuthors: Frank Nijhoff, Reinout Quispel, Sarah Lobb |
2008 | DIS |
Chebyshev polynomial approximations for some hypergeometric systemsAuthors: Juri Rappoport |
2008 | DIS |
How to detect the integrability of discrete systemsAuthors: Rod Halburd, Alfred Ramani, Claude Michel Viallet, B Grammaticos |
2008 | DIS |
Expansions on special solutions of the first $\it q$-Painlevé equation around the infinityAuthors: Yousuke Ohyama |
2008 | DIS |
The discrete potential Boussinesq equation and its multisoliton solutionsAuthors: Kenji Kajiwara, K Maruno |
2008 | DIS |
23 March 2009 to 27 March 2009
30 March 2009 to 3 April 2009
11 May 2009 to 15 May 2009
29 June 2009 to 3 July 2009
8 July 2013 to 12 July 2013
Tuesday 27th January 2009 | |||
---|---|---|---|
13:30 to 15:30 | Discussion Room |
Wednesday 28th January 2009 | |||
---|---|---|---|
14:00 to 15:00 |
Jon Nimmo University of Glasgow |
Room 1 |
Tuesday 3rd February 2009 | |||
---|---|---|---|
11:00 to 13:00 | Discussion Room | ||
14:00 to 15:00 | Discussion Room |
Wednesday 4th February 2009 | |||
---|---|---|---|
14:00 to 14:50 | Room 1 | |
|
15:10 to 16:00 |
Christian Korff University of Glasgow |
Room 1 |
Thursday 5th February 2009 | |||
---|---|---|---|
11:00 to 13:00 | Room 1 |
Wednesday 11th February 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Jarmo Hietarinta University of Turku |
Room 1 | |
15:10 to 15:50 |
Adam Doliwa University of Warmia and Mazury |
Room 1 | |
Thursday 12th February 2009 | |||
---|---|---|---|
11:00 to 12:00 |
Decio Levi Università degli Studi Roma Tre |
Room 1 | |
Wednesday 18th February 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Mourad Ismail University of Central Florida |
Room 1 | |
15:10 to 16:00 |
Yasuhiro Ohta Kobe University |
Room 1 |
Tuesday 24th February 2009 | |||
---|---|---|---|
11:00 to 13:00 |
Sergey Novikov University of Maryland, College Park |
Room 1 |
Wednesday 25th February 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Peter van der Kamp La Trobe University |
Room 1 | |
15:10 to 16:00 |
Rafael Hernandez Heredero E.U.I.T de Telecomunicaciones |
Room 1 | |
Wednesday 4th March 2009 | |||
---|---|---|---|
14:00 to 14:50 |
James Atkinson University of Sydney |
Room 1 | |
15:10 to 16:00 | Room 1 | |
Monday 9th March 2009 | |||
---|---|---|---|
17:00 to 18:00 |
Sergey Novikov University of Maryland, College Park |
Room 1 | |
Tuesday 10th March 2009 | |||
---|---|---|---|
11:00 to 12:30 |
Valentin Ovsienko Université Claude Bernard Lyon 1 |
Room 1 | |
Wednesday 11th March 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Pavlos Xenitidis University of Leeds |
Room 1 | |
15:10 to 16:00 |
Vassilios Papageorgiou University of Patras |
Room 1 | |
Tuesday 17th March 2009 | |||
---|---|---|---|
11:00 to 15:00 | Discussion Room |
Wednesday 18th March 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Alexander Rasin Bar-Ilan University |
Room 1 | |
15:00 to 16:00 |
Ian Marshall Loughborough University |
Room 1 | |
Thursday 19th March 2009 | |||
---|---|---|---|
11:00 to 11:50 |
Francesco Calogero Università degli Studi di Roma La Sapienza |
Room 1 | |
14:00 to 16:00 | Discussion Room |
Friday 20th March 2009 | |||
---|---|---|---|
14:00 to 15:00 |
Chris Ormerod La Trobe University |
Room 1 | |
Monday 30th March 2009 | |||
---|---|---|---|
09:00 to 17:00 | No Room Required |
Tuesday 31st March 2009 | |||
---|---|---|---|
09:00 to 17:00 | No Room Required |
Wednesday 1st April 2009 | |||
---|---|---|---|
09:00 to 17:00 | No Room Required |
Thursday 2nd April 2009 | |||
---|---|---|---|
09:00 to 17:00 | No Room Required |
Friday 3rd April 2009 | |||
---|---|---|---|
09:00 to 17:00 | No Room Required |
Wednesday 8th April 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Peter Ellsworth Hydon University of Surrey |
Room 1 | |
Wednesday 15th April 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Hjalmar Rosengren Chalmers University of Technology |
Room 1 | |
Wednesday 22nd April 2009 | |||
---|---|---|---|
14:00 to 14:50 | Room 1 | |
|
15:10 to 16:00 |
Peter Clarkson University of Kent |
Room 1 | |
Wednesday 29th April 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Christian Mercat Université de Montpellier 2 |
Room 1 | |
Wednesday 6th May 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Hidetaka Sakai University of Tokyo |
Room 1 | |
15:10 to 16:00 |
Elizabeth Mansfield University of Kent |
Room 1 | |
Thursday 7th May 2009 | |||
---|---|---|---|
11:00 to 13:00 |
Hans Duistermaat Universiteit Utrecht |
Room 1 | |
Saturday 9th May 2009 | |||
---|---|---|---|
18:00 to 19:00 | No Room Required |
Monday 18th May 2009 | |||
---|---|---|---|
17:00 to 18:00 |
Hans Duistermaat Universiteit Utrecht |
Room 1 | |
Tuesday 19th May 2009 | |||
---|---|---|---|
13:00 to 15:00 |
Hans Duistermaat Universiteit Utrecht |
Room 1 | |
Wednesday 20th May 2009 | |||
---|---|---|---|
10:00 to 12:30 |
Yang Shi University of Sydney |
Room 1 | |
14:00 to 14:50 |
Richard Askey University of Wisconsin-Madison |
Room 1 | |
15:10 to 16:00 |
Teruhisa Tsuda Kyushu University |
Room 1 |
Thursday 21st May 2009 | |||
---|---|---|---|
13:30 to 16:00 | Discussion Room |
Tuesday 26th May 2009 | |||
---|---|---|---|
13:00 to 13:50 |
John Roberts University of New South Wales |
Room 1 | |
14:00 to 14:50 |
Hans Duistermaat Universiteit Utrecht |
Room 1 | |
Wednesday 27th May 2009 | |||
---|---|---|---|
14:00 to 15:00 |
Peter Ellsworth Hydon University of Surrey |
Discussion Room |
Tuesday 2nd June 2009 | |||
---|---|---|---|
11:00 to 12:30 | Room 2 |
Wednesday 3rd June 2009 | |||
---|---|---|---|
11:00 to 12:30 | Room 1 | ||
14:00 to 14:50 |
Integrable self-adaptive mesh method for the Camassa-Holm and short pulse equations |
Room 1 |
Friday 5th June 2009 | |||
---|---|---|---|
14:00 to 16:00 |
Tomoyuki Takenawa Tokyo University of Marine Science and Technology |
Discussion Room |
Tuesday 9th June 2009 | |||
---|---|---|---|
11:00 to 12:00 |
Anne Granier Université Paul Sabatier Toulouse III |
Room 1 | |
Wednesday 10th June 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Igor Krichever Columbia University |
Room 1 | |
15:10 to 16:00 |
Boris Konopelchenko Universita' del Salento |
Room 1 | |
Friday 12th June 2009 | |||
---|---|---|---|
11:00 to 11:50 |
Generalised Kramers-Wannier duality for lattice spin models with non-abelian symmetry |
Room 1 | |
14:00 to 15:00 |
James Atkinson University of Sydney |
Room 2 |
Monday 15th June 2009 | |||
---|---|---|---|
11:00 to 12:30 | Room 1 |
Tuesday 16th June 2009 | |||
---|---|---|---|
11:00 to 11:50 |
Da-jun Zhang Shanghai University |
Room 1 | |
17:00 to 18:00 | Room 1 | |
Wednesday 17th June 2009 | |||
---|---|---|---|
14:00 to 14:50 |
Alexander Mikhailov University of Leeds |
Room 1 | |
15:10 to 16:00 |
Jean-Marie Maillard Université Pierre et Marie Curie Paris |
Room 1 | |
Thursday 18th June 2009 | |||
---|---|---|---|
11:00 to 12:00 |
Evgeny Korotyaev Cardiff University |
Room 1 | |
Friday 19th June 2009 | |||
---|---|---|---|
11:00 to 12:00 |
Boris Konopelchenko Universita' del Salento |
Room 2 |
Tuesday 23rd June 2009 | |||
---|---|---|---|
14:00 to 14:50 | Room 2 |
Wednesday 24th June 2009 | |||
---|---|---|---|
14:00 to 14:50 |
On the dynamics of the recurrence $x_{n + 1} = f(x _n)/x_{n-1}$ |
Room 2 | |
15:10 to 16:00 |
Alexander Veselov Loughborough University |
Room 1 | |
Thursday 25th June 2009 | |||
---|---|---|---|
14:00 to 15:00 | Room 2 |
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