Algebraic and analytic integral geometry

摘要奖：DMS 1406252，首席研究员：约瑟夫·H·G·傅成玉积分几何的学科是从计算随机形状或物体的统计数据的需要中产生的。基本思想是从分离某些有限相加的基本定量几何测量开始，从这个意义上说，如果一个物体是由两个较小的物体合并形成的，那么与总数相关的量等于与成分相关的量之和减去与它们重叠相关的量。这些量称为估值，包括三维物体的体积和表面积，或二维图形的面积和周长。关于给定估值如何在一系列相交对象上变化的统计数据被证明是由一套丰富的代数规则系统管理的，这构成了本项目的重点。因此，积分几何应该被视为整个几何的一个基本的重要方面，在科学上有许多可能的应用。所涉及的方法既属于代数，也属于分析。其目的是理解由估值的相互作用引起的代数问题，以及围绕其定义所必需的正则性的分析问题。在分析方向上，我们的主要目的是改进Pokorny和Rataj关于凸函数差可定义的集合的正则性的最新结果。在代数方面，过去十年取得了很大的进步，这在很大程度上是由S.Alesker在本世纪初引入的关于估值的基本新的代数结构所推动的。从这个角度来看，Blaschke，Santalo，Chern等人的经典积分几何，在其完全等距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微不足道的基本情况。第一种非平凡情形是厄米积分几何，即复空间形式的积分几何。我们最近的结果表明，与这类空间相关的代数非常丰富，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能力在这些情况下轻松地进行计算，以探索在没有这种广泛对称性的黎曼流形中估值是如何表现的。

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.