Special functions, dispersionless hierarchies, duality

在可交换环面和非可交换环面上由函数满足的方程将得到发展，特别注意它们的模性质和雅可比矩阵的分层。具体地说，Kleinian函数的模方面被提出与单变量多项式方程的解相联系（类似于超椭圆的tanulls在五次方程的Hermite解中的作用），并与等同构方程相联系，具有镜像对称的前景应用。模复制性质（hecke型作用）将与KP和Toda型的无色散层次联系起来进行研究，其中主要的开放问题是frobenius流形结构的构造，并具有在格大于1的曲线上的Hurwitz空间的解析理论的前景应用。博士后导师性质的相关项目（3属曲线及相关对偶的秩-2束模空间上的投影连接）将进行远程研究。特殊函数理论是通过数学和物理的强烈共生而产生的，这种共生渗透在19世纪分析和几何大师的工作中。20世纪70年代末，由于发现了偏微分方程的可积层次，数学和物理之间的联系出现了一次非同寻常的复兴。这引起了对模块化（函数对其控制的几何对象的依赖，例如曲线）的经典和新颖问题的强烈关注，这是本提案的主要焦点。模函数的分析将通过经典函数和量子函数在几何上应用于交换环面和非交换环面的分层。该领域的一个普遍特征是几何之间的“奇怪对偶性”：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，这一提议旨在给出一种新的量子可积性概念，从量子化函数的理论开始，理想地向物理模型的统一发展。学生和博士后将参与项目，并参加跨学科的研讨会和活动。

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.