Geometric Aspects of Integrable Systems

可积系统的几何方面

• 批准号: 0707132
• 负责人: Chuu-lian Terng
• 金额: $ 17.9万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707132&HistoricalAwards=false
• 关键词: Geometric; Aspects; Integrable; Systems

Equations for surfaces with constant negative curvature and isothermic surfaces in three space are soliton equations. Soliton theory have been applied successfully to study these geometric equations. Given a soliton equation, an interesting question is whether there is a geometric object whose controlling equation is the given soliton equation. Terng proposes to study the geometric aspects of the actions of Virasoro algebra, the submanifold geometry associated to soliton equations, the Hamiltonian theory of space-time monopole equations, and to finish a research monograph on integrable systems and differential geometry and a book on curves and surfaces.Soliton equations arise naturally in applied mathematics and differential geometry. For example, the non-linear Schrodinger equation (NLE) models the motion of the envelope of waves in optic fiber. Soliton waves for NLE have been used in telecommunications. Better understanding of soliton equations may provide more applications and interesting geometry. Terng has also contributed to the training of postdoctoral scholars, graduate students, and undergraduate students. She has also co-organized the Mentoring program for women mathematicians at IAS since 1994. She plans to continue these activities.

三空间中具有恒定负曲率的曲面和等温曲面的方程是孤子方程。 孤子理论已成功应用于研究这些几何方程。给定一个孤子方程，一个有趣的问题是是否存在一个几何对象，其控制方程是给定的孤子方程。 Terng 提议研究 Virasoro 代数作用的几何方面、与孤子方程相关的子流形几何、时空单极子方程的哈密顿理论，并完成一本关于可积系统和微分几何的研究专着以及一本关于曲线和曲面的书。 孤子方程自然出现在应用数学和微分几何中。例如，非线性薛定谔方程 (NLE) 对光纤中波包络的运动进行建模。 NLE 的孤子波已用于电信领域。更好地理解孤子方程可能会提供更多的应用和有趣的几何结构。 Terng 还为博士后学者、研究生和本科生的培养做出了贡献。自 1994 年以来，她还在 IAS 共同组织了针对女性数学家的指导计划。她计划继续这些活动。