Cluster algebras, Teichmüller theory and Macdonald polynomials

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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.