Surface Geometry and Integrable Systems: Theory and Experiment

三维空间中的曲面几何为各种数学领域的相互作用和交叉提供了一个自然的平台：曲面上的曲率关系产生了非线性椭圆偏微分方程，这也是更一般的非线性问题的范例;黎曼曲面理论，全纯向量丛（希格斯束），亚纯连接在特殊曲面类的研究中发挥着核心作用，例如恒定平均曲率（模型平衡流体界面）和Willmore表面（弹性膜）在空间形式，更一般地在调和映射的研究;特殊曲面的可积性方程通常可以用完全可积系统来表述; 2在低维中工作允许计算、可视化和实验来测试结构并获得意外现象的证据。在这方面，我们建议调查的结构和模空间的特殊曲面，主要是原型的情况下，常平均曲率曲面和调和映射到2-球面。

Surface geometry in 3-space provides a natural platform for the interaction and cross-fertilization of various mathematical areas: curvature relations on a surface give rise to non-linear elliptic PDEs which also are a paradigm for more general non-linear problems; Riemann surface theory, holomorphic vector bundles (Higgs bundles), and meromorphic connections play a central role in the study of special surface classes, such as constant mean curvature (model equilibrium fluid interfaces) and Willmore surfaces (elastic membranes) in space forms, and more generally in the study of harmonic maps; the integrability equations of special surfaces can often be rephrased in terms of completely integrable systems; working in low dimensions allows for computation, visualization and experimentation to test conjectures and obtain evidence of unexpected phenomena. In this context, we propose to investigate the structure and moduli spaces of special surfaces, mainly the prototypical case of constant mean curvature surfaces and harmonic maps into the 2-sphere.