Higher Order Problems in Geometric Analysis

几何分析中的高阶问题

• 批准号: EP/J004383/1
• 负责人: Roger Moser
• 金额: $ 2.2万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ004383%2F1
• 关键词: Higher; Order; Problems; Geometric; Analysis

Many problems in modern geometry are formulated in terms of nonlinear partial differential equations (PDEs), the analysis of which requires a high degree of expertise in the theory of PDEs as well as geometric insight. Since geometric principles or geometric constraints are often used in theoretical physics, engineering, or other sciences, the combination of analytic techniques with geometric ideas is also of tremendous interest outside of mathematics. Geometric analysis has thus always been interlinked with the theory of differential equations and with mathematical physics as well as geometry. But recently, interest in geometric PDEs has further increased through the work of Perelman, who solved a long-standing problem by proving the famous Poincare conjecture with a PDE-based approach, thereby increasing our understanding of three-dimensional spaces considerably.The bulk of existing work in the area is concerned with equations of order 2, but there is increasing interest in higher order problems. Typically these require completely new methods, because much of the second order theory relies heavily on the maximum principle, which is not available for higher order equations. We propose to hold a workshop on `Higher Order Problems in Geometric Analysis', bringing together some of the leading experts on problems of this sort. We envisage a meeting that not only allows an exchange of the latest ideas within the geometric analysis community, but also generates interactions with geometers, applied mathematicians, or engineers. Furthermore, PhD students and other young researchers should have the opportunity to learn about questions, ideas, and techniques that they may rarely encounter otherwise.

现代几何形状中的许多问题都是根据非线性偏微分方程（PDE）提出的，对PDE的理论以及几何见解，其分析需要高度的专业知识。由于几何原理或几何约束通常在理论物理，工程或其他科学中使用，因此在数学之外，分析技术与几何思想的结合也引起了极大的兴趣。因此，几何分析始终与微分方程理论以及数学物理学以及几何形状相互联系。但是最近，通过Perelman的工作，对几何PDE的兴趣进一步增加了，他们通过以基于PDE的方法证明著名的Poincare猜想来解决了一个长期存在的问题，从而增加了我们对三维空间的理解。通常，这些需要全新的方法，因为二阶理论的大部分时间都在很大程度上取决于最大原理，这对于高阶方程不可用。我们建议在“几何分析中的高阶问题”举办一个研讨会，将这种问题的一些领先专家汇总在一起。我们设想的会议不仅允许在几何分析社区中交换最新想法，还可以与几何学，应用数学家或工程师产生互动。此外，博士生和其他年轻研究人员应该有机会了解他们可能很少会遇到的问题，想法和技术。