Affine Invariants in Geometric Analysis

由于仿射几何在图像处理、信息技术和随机几何中的应用日益广泛，需要对凸体的仿射不变量进行新的研究。此外，还需要开发新技术来解决传统方法无法解决的长期问题。 我们的项目正好解决了这些问题上的仿射不变量的理解在这里大，包括中心仿射和equi-affine不变量。虽然我们的研究的一部分将采用新的曲率流来研究仿射不变量的凸体具有足够的规则的边界，我们将开始一个原始的研究新的仿射不变量的凸体缺乏光滑的正高斯曲率。长期目标是将这两类问题拟合成一个全局相干理论，该理论包括所有凸体，而不考虑特定的边界结构。 该项目的直接后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理论中产生的广义Minkowski问题的解决方案。一个特别的注意将给予开放的问题，在类的多面体和连接的理论价值。 为了增加项目的重要性，我们的方法将从凸，离散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主线。

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.