Special functions, dispersionless hierarchies, duality

特殊函数、无色散层次结构、对偶性

• 批准号: 0808708
• 负责人: Emma Previato
• 金额: $ 16.28万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808708&HistoricalAwards=false
• 关键词: Special; functions; dispersionless; hierarchies; duality

Equations satisfied by theta functions, on commutative and non-commutative tori, are to be developed, with special attention to their modular properties and to stratifications of the Jacobian theta divisor. Specifically, the modular aspect of Kleinian functions is proposed to be connected with the solution of polynomial equations in one variable (in analogy with the role of hyperelliptic thetanulls in Hermite's solution of the fifth-degree equation), and with equations of isomonodromy, with prospective applications to mirror symmetry. Modular reproducing properties (Hecke-type actions) are to be studied in connection with dispersionless hierarchies of KP and Toda type, for which the main open question is the construction of a Frobenius-manifold structure, with prospective applications to an analytic theory for Hurwitz spaces over curves of genus greater than one. Related projects of postdoctoral mentoring character (projective connection over moduli spaces of rank-2 bundles for curves of genus 3 and related dualities) are to be conducted long-distance.The theory of special functions came to life through the strong symbiosis of mathematics and physics that pervaded the work of the masters of analysis and geometry in the nineteenth century. An extraordinary revival of the connection between mathematics and physics came about in the late 1970s, triggered by the discovery of integrable hierarchies of partial differential equations. This gave rise to intense activity on both classical and novel questions of modularity (the dependence of a function on its governing geometric object, e.g., a curve), the main focus of this proposal. The analysis of modular functions is to be applied geometrically to stratifications of commutative and non-commutative tori, through classical and quantum theta functions. A widespread feature in this area is that of "strange dualities" between geometries: several modular interpretations are proposed, with consequent applications to systems of practical interest. In a nutshell, this proposal is aimed to giving a notion of quantum integrability of a new kind, starting with a theory of the quantized functions and ideally progressing toward unification of physical models. Students and postdoctoral fellows are to be involved in the projects and participate in workshops and activities of interdisciplinary kind.

将开发关于交换环和非交换环的 theta 函数满足的方程，特别注意它们的模属性和雅可比 theta 除数的分层。具体来说，克莱因函数的模方面被提出与一个变量的多项式方程的解有关（类似于超椭圆thetanulls在埃尔米特五次方程解中的作用），以及等单性方程，并具有镜像对称的预期应用。 将结合 KP 和 Toda 类型的无色散层次结构来研究模再现特性（Hecke 型作用），其中主要的悬而未决的问题是 Frobenius 流形结构的构造，并有望应用于大于 1 的亏格曲线上的 Hurwitz 空间的解析理论。 博士后指导性质的相关项目（3 列束的模空间上的射影连接，用于 3 型曲线和相关对偶性）将远程进行。特殊函数理论通过数学和物理的强烈共生而得以实现，这种共生贯穿了 19 世纪分析和几何大师的工作。 20 世纪 70 年代末，偏微分方程可积层次结构的发现引发了数学和物理学之间联系的非凡复兴。 这引发了关于模块化的经典和新颖问题（函数对其控制几何对象（例如曲线）的依赖性）的激烈活动，这是该提案的主要焦点。 模函数的分析将通过经典和量子 theta 函数在几何上应用于交换和非交换环面的分层。 该领域的一个普遍特征是几何形状之间的“奇怪的二元性”：提出了几种模块化解释，并随后应用于实际感兴趣的系统。 简而言之，该提议旨在给出一种新型量子可积性的概念，从量子化函数的理论开始，理想地朝着物理模型的统一迈进。 学生和博士后研究员将参与这些项目并参加跨学科的研讨会和活动。