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广泛应用，需要对凸体的新型仿射不变量进行新的研究。此外，需要开发新的技术来解决长期抵制传统方法的问题。我们的项目恰恰解决了这些问题，在这里广泛地理解为包括中心仿射和等仿射不变量的仿射不变量。虽然我们的部分研究将使用新的曲率流来研究边界足够规则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的原创性研究。长期目标是将这两类问题整合到一个全局一致的理论中，该理论包括所有的凸体，而不考虑特定的边界结构。该项目的直接结果将是仿射和等周不等式领域的新研究成果，以及在Brunn-Minkowski-Firey凸体理论中产生的广义Minkowski问题的解。我们将特别关注多面体类中的公开问题和与赋值理论的联系。除了这个项目的意义之外，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组和曲率流，同时遵循在某些变换组下不变的共同线索。

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.