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.

现代几何中的许多问题都是用非线性偏微分方程组(PDE)来描述的，对它的分析不仅需要几何洞察力，而且需要高度的偏微分方程组理论专业知识。由于几何原理或几何约束经常用于理论物理、工程或其他科学中，分析技术与几何思想的结合在数学之外也引起了极大的兴趣。因此，几何分析总是与微分方程式理论、数学物理和几何联系在一起。但最近，通过Perelman的工作，人们对几何偏微分方程组的兴趣进一步增加，他用基于偏微分方程组的方法证明了著名的Poincare猜想，从而解决了一个长期存在的问题，从而大大增加了我们对三维空间的理解。该领域现有的大部分工作涉及二阶方程，但对高阶问题的兴趣越来越大。这些通常需要全新的方法，因为许多二阶理论严重依赖于最大值原理，而这对于高阶方程是不可用的。我们建议举办一次关于“几何分析中的高阶问题”的研讨会，汇聚这类问题的一些顶尖专家。我们设想的会议不仅允许在几何分析社区内交流最新的想法，而且还可以与几何学家、应用数学家或工程师进行互动。此外，博士生和其他年轻的研究人员应该有机会了解他们在其他方面可能很少遇到的问题、想法和技术。