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

合作研究：Hurwitz 数、Teichmuller 空间、舒伯特微积分和簇代数

• 批准号: 0400484
• 负责人: Michael Gekhtman
• 金额: $ 10.6万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400484&HistoricalAwards=false
• 关键词: COLLABORATIVE; RESEARCH; Hurwitz; Numbers; Teichmuller

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.

该项目探讨了经典组合学、全纯曲线模空间理论、实代数几何和总正性以及新兴的簇代数理论之间的联系。特别是，我们计划利用装饰Teichmueller空间和簇代数理论之间的联系来研究模空间；开发簇代数的通用几何框架，并找到形式簇代数几何示例的足够通用的来源，以补充舒伯特簇产生的几何示例。此外，我们将结合几何Littlewood-Richardson规则的簇代数方法应用于真实舒伯特微积分中的Shapiro-Shapiro猜想。最后，我们计划推广 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都受到理论物理的启发，并通过不同分支之间的相互作用来实现数学，如组合学、几何（微分、辛和代数）、可积模型理论等等。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，福明和泽列文斯基最近发现的簇代数形式主义被证明特别适合研究物理上重要的坐标系。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中以及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有用。