Algebraic and analytic integral geometry

[摘要]奖励：DMS 1406252，首席研究员：Joseph H. G. fu。积分几何学科的发展源于对随机形状或物体进行统计计算的需要。其基本思想是首先分离出某些基本的定量几何测量，它们是有限相加的，也就是说，如果一个物体是由两个较小的物体合并而成的，那么与总数相关的量等于与组成部分相关的量的总和，减去与它们重叠相关的量。这样的量，称为估值，包括三维物体的体积和表面积，或二维图形的面积和周长。一个给定的估值如何在相交对象的家族中变化的统计结果是由一个丰富的代数规则系统控制的，这构成了这个项目的重点。因此，积分几何应该被看作是几何学的一个基本的重要方面，在科学上有许多可能的应用。所涉及的方法属于代数和分析。目标是理解由赋值的相互作用引起的代数问题，以及围绕其定义所需的正则性的分析问题。在解析方向上，我们的主要目的是改进Pokorny和Rataj最近关于凸函数差分可定义集合的正则性的结果。在代数方面，过去十年取得了很大的进展，主要是由S. Alesker在本世纪初提出的关于赋值的基本的新代数结构所催化的。从这个角度来看，Blaschke, Santalo， Chern等人的经典积分几何，在他们的全等长群的作用下处理真实的空间形式，似乎是一个更丰富的理论的平凡基础案例。第一种非平凡的情况是厄米积分几何，即复空间形式的积分几何。我们最近的结果表明，与这个空间族相关的代数是非常丰富的，具有许多神秘和无法解释的现象。我们的主要目标之一是利用我们新发现的在这些情况下进行计算的能力，在一定程度上探索没有如此广泛对称性的黎曼流形中的赋值行为。

AbstractAward: DMS 1406252, Principal Investigator: Joseph H. G. FuThe subject of integral geometry grows out of the need to calculate the statistics of random shapes or objects. The basic idea is to begin by isolating certain fundamental quantitative geometric measurements that are finitely additive, in the sense that if an object is formed by amalgamating two smaller objects then the quantity associated to the total is equal to the sum of the quantities associated to the constituents, minus the quantity associated to their overlap. Such quantities, called valuations, include the volume and the surface area of a three-dimensional object, or the area and the perimeter of a two-dimensional figure. The statistics of how a given valuation varies over families of intersecting objects turns out to be governed by a rich system of algebraic rules, which constitute the focus of this project. Integral geometry should thus be viewed as a fundamentally important aspect of geometry as a whole, with many possible applications to the sciences.The methods involved belong both to algebra and analysis. The objectives are to understand the algebraic questions arising from the interactions of valuations, and also the analytic problems surrounding the regularity properties necessary for their definition. In the analytic direction our main aim is to refine recent results of Pokorny and Rataj on the regularity of sets definable by differences of convex functions. On the algebraic side, the past decade has seen a great deal of progress, largely catalyzed by the fundamental new algebraic structures on valuations introduced by S. Alesker in the early years of the century. From this perspective the classical integral geometry of Blaschke, Santalo, Chern, et al., dealing with the real space forms under the actions of their full isometry groups, appears as the trivial ground case of a much richer theory. The first nontrivial case is that of Hermitian integral geometry, i.e. the integral geometry of the complex space forms. Our recent results reveal that the algebra associated with this family of spaces is extremely rich, characterized by many mysterious and unexplained phenomena. One of our primary goals is to exploit our new-found ability to make computations in these cases with some degree of ease to explore how valuations behave in Riemannian manifolds without such extensive symmetries.