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

这个项目探索了经典组合学、全纯曲线的模空间理论、实代数几何和全正性以及新兴的簇代数理论之间的联系。特别地，我们计划利用装饰的Teichmueller空间和簇代数理论之间的联系来研究模空间；建立簇代数的一般几何框架，并找到形式簇代数的几何例子的充分通用来源，以补充由Schubert簇产生的几何例子。此外，我们将把团簇代数方法与几何Littlewood-Richardson规则相结合，应用于实Schubert微积分中的Shapiro-Shapiro猜想。最后，我们计划推广关于Hurwitz数的ELSV公式，即将(双)Hurwitz数表示为模空间上某些特征类的积分。数学上的许多重大突破是受到理论物理的启发，并通过组合数学、几何(微分、辛和代数)、可积模型理论等不同数学分支之间的相互作用而实现的。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，Fomin和Zlevinsky最近发现的团簇代数形式主义被证明是唯一适合于研究物理上重要的坐标系的。扩展簇代数方法的范围将被证明在拓扑场论、二维引力、经典和量子可积模型以及在更适用的水平上在电子工程中，特别是在非线性滤波器的设计中是有用的。

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.