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.

该项目探讨了古典组合学，霍明态曲线模量空间的理论，实际代数几何和总阳性，以及群集代数的新出现的理论。特别是，我们计划使用装饰的Teichmueller空间与集群代数理论之间的联系来研究模量空间。为了开发一个用于群集代数的一般几何框架，并找到一个足够通用的形式示例的形式群集代数的几何示例，该群集补充了舒伯特品种产生的群集。此外，我们将应用群集代数方法与几何林木 - 里查森（Littlewood-Richardson）规则相结合，以在真实的舒伯特演算中的shapiro-shapiro猜想中。 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模特和许多其他。我们感兴趣的几何对象可用于描述物理系统的参数。在许多情况下，Fomin和Zelevinsky最近发现的集群代数形式主义事实非常适合研究物理上重要的坐标系统。扩展群集代数方法的范围将被证明在拓扑场理论，2-D重力，经典和量子整合模型中，并且在更适用的水平上，特别是在电气工程中，尤其是在非线性过滤器的设计中。