Integrable systems and applications

在其现代表述中（自“逆谱”方法的发现以来），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点阵模型）；3)随机矩阵的谱理论（应用于大核谱的统计理论；与黎曼曲面的模空间有关的图解枚举问题，与随机算子的特征值和离散随机过程有关的“普适”现象；4）枚举几何、模空间和组合学（Gromov-Witten和Donaldson-Thomas不变量；Hurwitz数，Hodge不变量；拓扑递推）；5)随机生长过程和可积概率（晶体生长、不相容过程、Schur过程）；6)Riemann曲面上亚纯协变导数的等单调变形。一个关键因素是“Tau函数”的概念，正如佐藤等人所介绍的那样。这些可以被不同地看作：1)在正则变换理论的意义上，生成一组完备交换流的函数；2)等形变动力学生成函数；3)随机矩阵模型族相对于参数测度族的配分函数和多点相关器；4)“可积”随机过程随机动力学的转移概率生成函数；5)在组合学的意义上，生成上述各种几何、枚举、几何和拓扑不变量的函数。本提案旨在进一步发展作者在可积系统理论中所介绍的一些关键概念和方法，如前所述。即，我们建议：1。在tau函数和可积系统的框架内进一步发展了“加权Hurwitz数”及其生成函数的概念，并将其嵌入到Eynard和Orantin的拓扑递归框架中。（与埃纳德等人合作）2 .研究了“量子Hurwitz数”的半经典渐近性和小参数极限。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Grassmannians和tau函数的框架下分析多面体递推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.