Affine Invariants in Geometric Analysis

鉴于仿射几何在图像处理、信息技术和随机几何中的应用爆炸式增长，需要对凸体进行新的仿射不变量研究。此外，有必要开发新技术来解决传统方法无法解决的长期问题。我们的项目精确地解决了这些关于仿射不变量的问题，这里大致理解为包括中心仿射和等仿射不变量。虽然我们的部分研究将采用新的曲率流来研究具有足够规则边界的凸体的仿射不变量，但我们将开始对缺乏光滑正高斯曲率的凸体的新仿射不变量进行原创性研究。长期目标是将这两类问题拟合成一个包括所有凸体而不考虑特定边界结构的全局相干理论。该项目的直接成果将是仿射和等周不等式领域的新研究成果，以及凸体的Brunn-Minkowski-Firey理论中出现的广义Minkowski问题的解决方案。将特别关注多面体类中的开放问题以及与估值理论的联系。为了增加项目的重要性，我们的方法将从凸几何、离散几何和微分几何到偏微分方程和曲率流，同时遵循某些变换组下的不变性的共同线索。

Given the explosion of applications of affine geometry to image processing, information technology and stochastic geometry, new research is needed to investigate novel affine invariants for convex bodies. Additionally, there is a need to develop new techniques to tackle long-standing problems which resisted conventional methods.Our project addresses precisely these issues on the subject of affine invariants understood here at large as to include centro-affine and equi-affine invariants. While part of our study will employ new curvature flows to study affine invariants for convex bodies with sufficiently regular boundary, we will start an original study of new affine invariants for convex bodies which are lacking smooth positive Gauss curvature. The long-term goal is fitting the two classes of problems into a global coherent theory which includes all convex bodies disregarding a particular boundary structure.The direct consequences of the proposed project will be novel research results in the area of affine and isoperimetric inequalities, and solutions to generalized Minkowski problems arising in the Brunn-Minkowski-Firey theory of convex bodies. A particular attention will be given to open problems in the class of polytopes and connections to the theory of valuations.Adding to the significance of the project, our methods will range from convex, discrete and differential geometry to partial differential equations and curvature flows, while following the common thread of invariance under certain groups of transformations.