## Symmetries, Integrability and Discrete Calculus

At the most fundamental level, the universe seems to be made of discrete things rather than continuous stuff, but calculus models continuous behaviour. Better mathematical tools are needed to handle discrete events. Symmetries and Integrability are among them.

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Space-time is quantized, characterised by the Planck length. Even on larger scales, where the world appears continuous, many important phenomena are discrete, such as those occurring in crystals or in molecular or atomic chains. Thus, difference equations may be more fundamental than differential ones. Moreover, differential equations often have to be solved numerically and that means that they have to be discretised – that is, approximated by a difference system.

The SPEDIS (Symmetry preserving discretization of integrable, superintegrable and nonintegrable systems) project was to develop and apply efficient mathematical tools for studying quantum and classical phenomena in a discrete setting.

Researchers' main interest was in models that can have symmetry and integrability properties, in particular finite and infinite dimensional integrable and superintegrable models. Integrable systems have as many commuting integrals of motion as degrees of freedom (which may be infinite). Superintegrable systems have more integrals of motion than degrees of freedom, and these integrals form interesting non-Abelian algebras. The integrals of motion are related to symmetries of the system. These may be Lie point symmetries, but usually they are generalised symmetries and they form more general algebras.

The objective of the project was to study and use Lie symmetries of difference equations and to discretise differential equations, preserving their most important properties. These include their Lie point symmetries, generalised symmetries, integrability and superintegrability.

Project work produced many scientific papers. Four publications are devoted to a new field, namely the symmetry preserving discretisation of partial differential equations, and have practical applications in the field of geometrical integration. Three concern equations with infinite dimensional symmetry algebras. Three others concern non-linear ordinary differential and difference equations: the construction of first integrals, discretisation and non-linear superposition formulas. Four are devoted to new types of superintegrable systems. They involve particles in magnetic fields, particles with spin, and with higher-order integrals of motion.

The SPEDIS (Symmetry preserving discretization of integrable, superintegrable and nonintegrable systems) project was to develop and apply efficient mathematical tools for studying quantum and classical phenomena in a discrete setting.

Researchers' main interest was in models that can have symmetry and integrability properties, in particular finite and infinite dimensional integrable and superintegrable models. Integrable systems have as many commuting integrals of motion as degrees of freedom (which may be infinite). Superintegrable systems have more integrals of motion than degrees of freedom, and these integrals form interesting non-Abelian algebras. The integrals of motion are related to symmetries of the system. These may be Lie point symmetries, but usually they are generalised symmetries and they form more general algebras.

The objective of the project was to study and use Lie symmetries of difference equations and to discretise differential equations, preserving their most important properties. These include their Lie point symmetries, generalised symmetries, integrability and superintegrability.

Project work produced many scientific papers. Four publications are devoted to a new field, namely the symmetry preserving discretisation of partial differential equations, and have practical applications in the field of geometrical integration. Three concern equations with infinite dimensional symmetry algebras. Three others concern non-linear ordinary differential and difference equations: the construction of first integrals, discretisation and non-linear superposition formulas. Four are devoted to new types of superintegrable systems. They involve particles in magnetic fields, particles with spin, and with higher-order integrals of motion.