Cluster algebras, Teichmüller theory and Macdonald polynomials

这是一个跨学科的建议，结合了表示论，几何和拓扑，微分和q-差分方程的技术，使代数和可积系统的明确和新颖的进展。特别是，该项目旨在建立一个集群代数理论和麦克唐纳多项式之间的桥梁。集群代数是最令人兴奋的数学最近的发明之一。在他们发现之后不久，由于Fomin和Zelevinsky在2002年，事实证明，簇代数与许多其他领域有关，例如物理学中的热力学Bethe Anastomics，多面体的组合学，泊松几何，以及与当前项目更相关的Teichmüller空间在几何和量子引力中的描述。这些联系汇集了来自数学和数学物理的许多不同分支的研究人员，这导致了簇代数理论和相关领域的迅速发展。对称函数在数学的许多领域发挥着关键作用，包括多项式方程理论，有限群表示理论，李代数，代数几何和特殊函数理论。麦克唐纳多项式是与仿射根系相关的多变量正交多项式族。这些多项式包含大多数以前研究的对称函数族作为特殊情况，并满足许多令人兴奋的组合性质。这个项目将开辟数学研究的新领域。事实上，当两个丰富的数学分支被统一起来时，总是会出现许多有趣的新问题，并会证明许多意想不到的结果。

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.