Cluster algebras, Teichmüller theory and Macdonald polynomials

这是一项跨学科的提议，结合了表示论、几何和拓扑学、微分和Q-差分方程的技术，在代数和可积系统方面取得了决定性的和新的进展。特别是，该项目的目标是在簇代数理论和麦克唐纳多项式之间建立一座桥梁。簇代数是数学中最令人兴奋的最新发明之一。在他们发现后不久，由于Fomin和Zlevinsky在2002年的发现，结果证明团簇代数与许多其他领域有关，例如物理学中的热力学Bethe Anatz，多面体的组合学，泊松几何，以及更相关的，几何和量子引力中的Teichmüler空间的描述。这些联系汇聚了来自数学和数学物理许多不同分支的研究人员，导致了簇代数理论和相关领域的惊人快速发展。对称函数在许多数学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。Macdonald多项式是与仿射根系有关的多个变量的一族正交多项式。这些多项式包含了大部分已有研究的对称函数族作为特例，并且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，当数学的两个丰富的分支被统一起来时，就会出现许多有趣的新问题，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.