Special functions, dispersionless hierarchies, duality

由交换环面和非交换环面上的theta函数满足的方程将被建立，并特别注意它们的模性质和Jacobian theta除数的分层。具体地说，Kleian函数的模方面被建议与一元多项式方程的解相联系(类似于超椭圆方程在Hermite解五次方程中的作用)，并与等单行方程相联系，具有镜像对称性的潜在应用。模再生性质(Hecke型作用)将结合Kp和Toda型无色散族来研究，其主要开放问题是构造Frobenius-流形结构，并有望应用于亏格大于1的曲线上的Hurwitz空间的分析理论。有关博士后导师特征(3亏格曲线和相关对偶曲线在2阶丛的模空间上的射影连接)的相关项目将远距离进行。特殊函数理论是通过19世纪分析和几何大师的工作中渗透的数学和物理的强烈共生而变得生动起来的。20世纪70年代末，由于偏微分方程系的发现，数学和物理之间的联系出现了非同寻常的复兴。这引发了对模块化(函数对其支配几何对象的依赖性，例如曲线)这一经典和新问题的强烈关注，这是本提案的主要焦点。模函数的分析将通过经典和量子theta函数几何地应用于交换环面和非交换环面的分层。这一领域的一个普遍特征是几何之间的“奇怪的对偶性”：提出了几种模数解释，并由此应用于具有实际意义的系统。简而言之，这个提议旨在给出一种新的量子可积性的概念，从量子化函数理论开始，理想地朝着物理模型的统一发展。学生和博士后研究员将参与这些项目，并参加跨学科性质的讲习班和活动。

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.