Cluster algebras, Teichmüller theory and Macdonald polynomials

This is a cross-disciplinary proposal combining techniques from representation theory, geometry and topology, differential and q-difference equations to make definitive and novel progress in algebra and integrable systems. In particular the project aims to build a bridge between cluster algebra theory and Macdonald polynomials.

Cluster algebras are one of the most exciting recent inventions in mathematics. Soon after their discovery, due to Fomin and Zelevinsky in 2002, it turned out that cluster algebras are connected to many other fields, such as the thermodynamic Bethe Ansatz in physics, combinatorics of polytopes, Poisson geometry and, more relevantly for this current project, the description of Teichmüller spaces in geometry and quantum gravity.
These connections brought together researchers from many different branches of mathematics and mathematical physics, which induced amasingly rapid growth both of the theory of cluster algebras and of related fields.

Symmetric functions play a key role in many areas of mathematics including the theory of polynomial equations, representation theory of finite groups, Lie algebras, algebraic geometry, and the theory of special functions. Macdonald polynomials are a family of orthogonal polynomials in several variables associated with affine root systems. These polynomials contain most of the previously studied families of symmetric functions as special cases, and satisfy many exciting combinatorial properties.

This project will open up new lines of research in mathematics. In fact, it is always the case that when two rich branches of mathematics are unified, many interesting new questions will arise and many unexpected result will be proved.

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 results will indeed create an economic or social impact. For example the use of number theory in modern cryptography, or the use 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 famous Jaffe report (available on the internet).

However it is true that often fundamental research which has a wide Academic impact will eventually have an even wider impact on 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".

The short term impact of this project is educational. In fact some of the basic ideas in cluster algebras are fairly easy to explain to secondary school students who could be involved in drawing collections of non-intersecting arcs on a non-compact Riemann surface with at least one cusp on the boundary. They could then compare different systems of arcs on the same surface and try to produce a transformation from one system to the other.
I have requested funds within this research proposal to run an outreach activity at Loughborough University aimed at year 12 students. The idea is to host them for two days to work on a research project on these topics.