Higher Order Problems in Geometric Analysis

现代几何中的许多问题都是用非线性偏微分方程（PDE）来表述的，对这些问题的分析需要高度的偏微分方程理论和几何洞察力。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，因此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理论、数学物理学以及几何学联系在一起。但是最近，Perelman的工作使人们对几何偏微分方程的兴趣进一步增加，他用基于偏微分方程的方法证明了著名的Poincare猜想，解决了一个长期存在的问题，从而大大提高了我们对三维空间的理解。在这个领域中，现有的大部分工作都是关于二阶方程的，但是对高阶问题的兴趣越来越大。通常这些需要全新的方法，因为大部分的二阶理论严重依赖于最大值原理，这是不适用于高阶方程。我们建议举办一个关于“几何分析中的高阶问题”的讲习班，将这类问题的一些主要专家聚集在一起。我们设想的会议，不仅允许几何分析社区内的最新思想交流，而且还产生与几何，应用数学家，或工程师的互动。此外，博士生和其他年轻的研究人员应该有机会了解他们可能很少遇到的问题，想法和技术。

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.