Integrable systems and applications

可集成系统和应用

• 批准号: RGPIN-2017-04805
• 负责人: Harnad, John
• 金额: $ 1.75万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679795
• 关键词: Integrable; systems; applications

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.*****

在其现代表述中（自从发现“逆谱”方法以来），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) 随机矩阵谱理论（应用于大核谱统计理论；与黎曼曲面模空间相关的图解枚举问题，以及关于随机算子特征值和离散随机过程的“普遍性”现象；4 ）枚举几何、模空间和组合（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) 黎曼曲面上亚纯协变导数的等单向变形。*** 一个关键要素是“Tau 函数”的概念，如由佐藤等人介绍。这些可以被不同地视为： 1）在规范变换理论意义上，一组完整的通勤流的生成函数； 2) 等单向变形动力学的生成函数； 3) 随机矩阵模型族的配分函数和多点相关器，相对于参数测量族； 4) 生成“可积”随机过程的随机动力学基础的转移概率函数； 5) 组合学意义上的上述各种几何、枚举、几何和拓扑不变量的生成函数。 *** 本提案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”的概念及其生成函数，并将其嵌入 Eynard 和 Orantin 的拓扑递归框架中。 （与 Eynard 等人合作）。***2。研究“量子赫尔维茨数”的半经典渐近和小参数极限（作者最近介绍）。***3。将“簇突变”生成的离散可积动力学分析为 Lax 矩阵的等谱流，并分析各向同性格拉斯曼函数和 tau 函数框架中的多面体递归关系生成的离散可积动力学。*****