Affine geometric analysis

在一系列的论文中，PI和她的合作者开始了仿射几何分析中的赋值或可加性测度的系统研究。在欧几里德几何中，赋值的概念早已被认为具有根本的重要性。一个具有里程碑意义的结果是哈德维格的分类刚性运动不变，连续估值凸机构于1950年。然而，经典的仿射表面积和中心仿射表面积是不连续的，因此不具有哈德维格定理的特征。在联合文件中，PI和M。Reitzner得到了仿射不变量、上连续赋值的分类，并建立了仿射表面积和中心仿射表面积的第一个刻画。该研究的主要目标是完成仿射赋值的分类，并使用为这种分类开发的结果和技术来解决仿射几何分析中的基本问题。基本对象是普通欧几里得空间中的函数和算子，它们在线性变换下是不变或协变的。在过去的几年里，大量的研究致力于深入调查这些功能和运营商之间的关系。有许多应用这些结果和拟议的研究的渐近理论的Banach空间，微分几何，常微分方程和偏微分方程，甚至看似无关的领域，如几何层析成像，图像处理和信息论。

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.