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

本课题探讨经典组合学、全纯曲线的模空间理论、实代数几何和全正性理论以及新兴的簇代数理论之间的联系。特别地，我们计划利用修饰的Teichmueller空间和簇代数理论之间的联系来研究模空间；发展聚类代数的一般几何框架，并找到形式聚类代数的几何实例的足够一般的来源，以补充由舒伯特变体产生的那些。此外，我们将结合几何Littlewood-Richardson规则的聚类代数方法应用于实际Schubert微积分中的Shapiro-Shapiro猜想。最后，我们计划推广赫尔维茨数的elsv公式，即将（二重）赫尔维茨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受到理论物理学的启发，并通过数学不同分支之间的相互作用实现，如组合学、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发现的聚类代数形式被证明是唯一适合于研究物理上重要的坐标系的方法。扩展聚类代数方法的范围将在拓扑场论、二维重力、经典和量子可积模型中被证明是有用的，并且在更适用的层面上，在电气工程中，特别是在非线性滤波器的设计中。

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.