Integrable systems and applications

在它的现代表述中(自从“逆谱”方法被发现以来)，可积系统理论在数学物理和纯数学的各个领域都产生了巨大的影响，包括：1)非线性可积动力学(孤子和非线性准周期流，应用于光学、流体动力学、超导现象)；2)量子逆散射方法(应用于统计力学中的量子自旋链、顶点模型和其他可积晶格模型)；3)随机矩阵的谱理论(应用于大核光谱的统计理论)；与Riemann曲面的模空间有关的图形计数问题，以及关于随机算子的特征值和离散随机过程的“普适性”现象；4)计数几何、模空间和组合学(Gromov-Witten和Donaldson-Thomas不变量；Hurwitz数，Hodge不变量；拓扑递归)；*5)随机生长过程和可积概率(晶体生长、排除过程、Schur过程)和6)Riemann曲面上亚纯协变导数的等同向变形。*一个关键元素是Sato等人引入的“Tau函数”的概念。它们可以不同地被看作：1)正则变换理论意义上的生成函数，一套完整的通勤流量的生成函数；2)同名变形动力学的生成函数；3)关于参数度量族的随机矩阵模型族的配分函数和多点相关器；4)用于作为“可积”随机过程的随机动力学基础的转移概率的生成函数；5)上述各种几何、计数、几何和拓扑不变量的组合意义上的生成函数。*本建议旨在进一步发展作者在可积系统理论中介绍的一些关键概念和方法，如上所述。也就是说，我们建议：*1.在tau函数和可积系统的框架内进一步发展“加权Hurwitz数”及其母函数的概念，并将其嵌入到Eynard和Orantin的拓扑递推框架中。(与Eynard等人合作)。*2.研究“量子Hurwitz数”的半经典渐近性和小参数极限(作者最近介绍)。*3.在各向同性Grassmannians和tau函数的框架下，分析由“簇突变”产生的离散可积动力学作为Lax矩阵的等谱流，并分析由多重递归关系产生的离散可积动力学。

In its modern formulation (since the discovery of the "inverse spectral" method), the theory of integrable systems has had enormous impact in a variety of domains, both in mathematical physics and pure mathematics, including: 1) nonlinear integrable dynamics (solitons and nonlinear quasi-periodic flows, with applications to: optics, fluid dynamics, superconductivity phenomena); 2) the quantum inverse scattering method (with applications to quantum spin chains, vertex models and other integrable lattice models in statistical mechanics); 3) the spectral theory of random matrices (with applications to the statistical theory of spectra of large nuclei; graphical enumeration problems relating to moduli spaces of Riemann surfaces, and "universality" phenomena regarding the eigenvalues of random operators, and discrete random processes; 4) enumerative geometry, moduli spaces and combinatorics (Gromov-Witten and Donaldson-Thomas invariants; Hurwitz numbers, Hodge invariants; Topological Recursion); ***5) random growth processes and integrable probabiity (crystal growth, exclusion processes, Schur processes) and 6) Isomonodromic deformations of meromorphic covariant derivatives on Riemann surfaces.*** A key element is the notion of "Tau functions", as introduced by Sato et al. These may be seen, variously, as: 1) generating functions, in the sense of canonical transformation theory, of a complete set of commuting flows; 2) generating functions for isomonodromic deformations dynamics; 3) partition functions and multipoint correlators for families of random matrix models, with respect to parametric families of measures; 4) generating functions for transition probabilities underlying random dynamics of "integrable" random processes; 5) generating functions, in the sense of combinatorics, of the various geometric, enumerative, geometrical and topological invariant mentioned above. *** This proposal aims at the further development of some key concepts and methods introduced by the author in the theory of integrable systems, as discussed above. Namely, we propose to:***1. Develop further the notion of "weighted Hurwitz numbers" and their generating functions within the framework of tau functions and integrable systems and embed this within the framework of Topological Recursions of Eynard and Orantin. (In collaboration with Eynard, and others).***2. Study the semiclassical asymptotics and small parameter limits of "Quantum Hurwitz numbers" (recently introduced by the author).***3. Analyze the discrete integrable dynamics generated by "cluster mutations" as isospectral flows of Lax matrices and to analyze the discrete integrable dynamics generated by polytope recursion relations in the framework of isotropic Grassmannians and tau functions.*****