Integrable systems and applications

在其现代表述中，自“逆谱”方法发现以来，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，其中包括：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（在量子自旋链、顶点模型和统计力学中的其他可积格模型中的应用）; 3）随机矩阵的谱理论（附对大原子核谱统计理论的应用;图形计数问题有关的模空间的黎曼曲面，和“普遍性”的现象，关于本征值的随机运营商，和离散随机过程; 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，排斥过程，Schur过程）和6）Riemann曲面上亚纯协变导数的Isomonodromic变形。 一个关键要素是“Tau函数”的概念，如Sato等人所介绍的。这些函数可以被不同地看作：1）在正则变换理论的意义上，一组完整的交换流的生成函数; 2）等单道形变动力学的生成函数; 3）随机矩阵模型族关于参数测度族的配分函数和多点算子; 4）“可积的”随机过程的随机动力学基础的转移概率的生成函数; 5）在组合学的意义上，上述各种几何、枚举、几何和拓扑不变量的生成函数。*** 这一建议旨在进一步发展作者在可积系统理论中引入的一些关键概念和方法，如上所述。即，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”及其生成函数的概念，并将其嵌入Eynard和Orantin的拓扑递归框架内。(In与Eynard和其他人合作）。2.研究“量子赫维茨数”（作者最近引入）的半经典渐近性和小参数极限。* 3.将“簇突变”产生的离散可积动力学分析为Lax矩阵的等谱流，并在各向同性Grassmannian和tau函数的框架下分析由多面体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.*****