COLLABORATIVE RESEARCH: Hurwitz Numbers, Teichmueller Spaces, Schubert Calculus and Cluster Algebras

本计画探讨古典组合学、全纯曲线的模空间理论、真实的代数几何与全正性，以及新兴的群代数理论之间的关联。特别是，我们计划使用之间的联系装饰Teichmueller空间和理论的集群代数调查模空间，开发一个一般的几何框架集群代数，并找到一个足够通用的来源几何例子正式集群代数补充所产生的舒伯特品种。此外，我们将应用集群代数方法结合几何Littlewood-Richardson规则的Shapiro-Shapiro猜想在真实的舒伯特演算。最后，我们计划推广Hurwitz数的ELSV-公式，即把（二重）Hurwitz数表示为模空间上某些特征类的积分。数学中的许多重大突破都是受到理论物理的启发，并通过不同数学分支之间的相互作用而实现的，如组合数学、几何（微分、辛和代数）、可积模型理论等。我们感兴趣的几何对象可以用来描述物理系统的参数。在许多情况下，最近由Fomin和Zelevinsky发现的簇代数形式主义被证明是唯一适合于物理上重要的坐标系的研究。扩大集群代数方法的范围将证明是有用的拓扑场论，2-D重力，经典和量子可积模型，并在更适用的水平，在电气工程，特别是在非线性滤波器的设计。

This project explores links between classical combinatorics, theory of moduli spaces of holomorphic curves, real algebraic geometry and total positivity, and the newly emerging theory of cluster algebras. In particular,we plan to use the link between decorated Teichmueller spaces and theory of cluster algebras to investigate moduli spaces; to develop a general geometric framework for cluster algebras and to find a sufficiently generic source of geometric examples of formal cluster algebras supplementing those arising from Schubert varieties. In addition, we will apply the cluster algebra approach combined with a geometric Littlewood-Richardson rule to the Shapiro-Shapiro conjecture in the real Schubert calculus. Finally, we plan to generalize ELSV-formula for Hurwitz numbers, namely, to express (double) Hurwitz numbers as integrals of certain characteristic classes over moduli spaces.Many significant breakthroughs in mathematics are inspired by theoretical physics and achieved through the interaction between different branches of mathematics such as combinatorics, geometry (differential, symplectic and algebraic), the theory of integrable models, and many others. Geometric objects we are interested in can be used to describe parameters of physical systems. In many cases, the cluster algebra formalism, recently discovered by Fomin and Zelevinsky, turns out to be uniquely suited for an investigation of physically important coordinate systems. Extending the scope of the cluster algebra approach will prove useful in topological field theory, 2-D gravity, classical and quantum integrable models and, on a more applicable level, in electrical engineering, in particular, in the design of nonlinear filters.