Affine geometric analysis

在一系列的论文中，PI和她的合作者开始系统地研究仿射几何分析中的估值或有限加性测度。在欧几里得几何中，估值的概念一直被认为是至关重要的。一个具有里程碑意义的结果是1950年哈德维格对凸体的刚性运动不变连续估值的分类。然而，经典仿射表面积和中心仿射表面积是不连续的，因此不能用哈德维格定理来表征。在联合论文中，PI和M. Reitzner获得了仿射不变量、上半连续值的分类，并首次建立了仿射表面积和中心仿射表面积的表征。本研究的主要目标是完成仿射估值的分类，并利用该分类的结果和技术来解决仿射几何分析中的基本问题。该建议集中在仿射几何分析的基本开放问题。基本对象是普通欧几里德空间中在线性变换下不变或协变的函数和算子。在过去的几年中，大量的研究致力于深入研究这些函数和运算符之间的关系。这些结果和提出的研究在Banach空间的渐近理论、微分几何、常微分方程和偏微分方程，甚至在几何层析成像、图像处理和信息论等看似无关的领域都有许多应用。

AbstractIn a series of papers, the PI and her collaborators started the systematic study of valuations or finitely additive measures within affine geometric analysis. In Euclidean geometry, the concept of valuation has long been known to be of fundamental importance. A landmark result was Hadwiger's classification of rigid motion invariant, continuous valuations on convex bodies in 1950. However, classical affine surface area and centro-affine surface area are not continuous and therefore not characterized by Hadwiger's theorem. In joint papers, the PI and M. Reitzner obtained a classification of affine invariant, upper semicontinuous valuations and established the first characterization of affine surface area and centro-affine surface area. The main goal of the proposed research is to complete the classification of affine valuations and use the results and techniques developed for this classification to solve fundamental problems in affine geometric analysis.The proposal is centered on fundamental open questions in affine geometric analysis. The basic objects are functions and operators in ordinary Euclidean space that are invariant or covariant under linear transformations. Within the last few years, a substantial amount of research was devoted to investigate in depth the relations between these functions and operators. There are numerous applications of these results and the proposed research to the asymptotic theory of Banach spaces, differential geometry, ordinary and partial differential equations, and even to seemingly unrelated fields like geometric tomography, image processing and information theory.