Special functions, dispersionless hierarchies, duality

满足θ函数方程，交换和非交换环，将开发，特别注意其模块化的性能和分层的雅可比θ因子。具体来说，克莱因函数的模块化方面被建议与一个变量的多项式方程的解（类似于超椭圆thetanulls在Hermite的五次方程的解中的作用），以及与isomonodromy方程，具有镜像对称的应用前景。 模再生性质（Hecke型作用）将结合KP型和户田型的无色散层次进行研究，其中主要的开放问题是构造Frobenius流形结构，并将应用于亏格大于1的曲线上的Hurwitz空间的分析理论。 相关项目的博士后导师字符（射影连接模空间的秩-2丛曲线的亏格3和相关的对偶）是进行远程。理论的特殊功能来生活通过强大的共生数学和物理学的工作，弥漫在硕士分析和几何世纪。 20世纪70年代末，数学和物理之间的联系出现了非凡的复兴，这是由偏微分方程的可积层次的发现引发的。 这引起了对经典和新颖的模块性问题（函数对其支配几何对象的依赖性，例如，曲线），这是该提案的主要焦点。 模函数的分析是几何适用于分层的交换和非交换环面，通过经典和量子θ函数。 在这一领域的一个普遍特征是几何之间的“奇怪的对偶性”：提出了几个模块化的解释，从而应用于实际感兴趣的系统。 简而言之，这个提议旨在给出一种新的量子可积性概念，从量子化函数理论开始，理想地朝着统一物理模型的方向发展。 学生和博士后研究员将参与项目，并参加跨学科的讲习班和活动。

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.