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解计数问题，以及与随机算子和离散随机过程的特征值有关的“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；拓扑递推)；

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);