Poisson Algebras of Holonomy Functions on Riemann Surfaces

This is a project in Pure Mathematics (Integrable Systems), to attract two outstanding scientist, Prof. L. Chekhov, for one year and Prof. Jorgen Andersen for a total period of one month to the Mathematics Department at Loughborough University.

Classically, physical phenomena are generally described by differential equations or, in other words, by equations which involve certain physical quantities (such as the position of a particle) and their variations (such as the particle velocity or its acceleration). Usually differential equations are very difficult or impossible to solve. Nevertheless there is a special class of differential equations (called integrable), which can be rewritten in the Lax form and therefore can be interpreted as an isospectral deformation. When we have a Lax representation for a physical system, then we can use many beautiful mathematical tools to understand, and often predict, its behaviour. In this project we will concentrate on a special class of equations which admit Lax representation: the so called Isomonodromic Deformations.

In particular we will construct an isomonodromic deformation which will be related to a certain abstract algebra. Algebras of this kind give the correct set up for quantisation. Indeed, at quantum level the physical quantities are replaced by operators called observables belonging to some abstract algebras. For this reason the study of such algebras has many applications in Applied Mathematics and Theoretical Physics.

Finally, we will give a geometric characterisation for this algebra, based on the celebrated Goldman bracket. This will allow us to establish a link between our work and the filed of Algebraic Geometry in Pure Mathematics.

Research in fundamental sciences such as pure mathematics takes a very long time to impact on society at large and as a consequence it is almost impossible to foresee if and how a certain mathematical result will indeed create an economic or social impact. For example the use of number theory in modern cryptography, or the use the differential geometry in GPS systems, were not foreseen at the time in which these theories were developed more than eighty years ago. These two examples are typical as shown in the Jaffe report \cite{Ja}.

However it is true that often fundamental research which has a wide Academic impact will eventually have a even wider impact into society. For example the work by Hardy in number theory in the nineteen-thirties was certainly acclaimed by his contemporary academic community, despite the fact that nobody could then foresee its applications in modern cryptography.

For this reason it is important at this stage that the proposed research is broad in its Academic beneficiaries and that a good dissemination strategy is in place. Both these points are addressed in the attached document called "Pathways to impact".