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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635584
• 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);

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产生了巨大影响，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（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）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;拓扑递归）;