Algebraic and analytic integral geometry

摘要奖：DMS 1406252，主要研究者：Joseph H. G.积分几何的主题是出于计算随机形状或物体的统计数据的需要。其基本思想是开始时，先将某些基本的定量几何测量分离出来，这些测量是可加性的，也就是说，如果一个物体是由两个较小的物体合并而成的，那么与总和相关的量等于与组成部分相关的量之和，减去与它们的重叠相关的量。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。一个给定的估值如何在相交对象的家庭中变化的统计结果是由一个丰富的代数规则系统所支配的，这构成了这个项目的重点。因此，积分几何应该被看作是整个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，所涉及的方法既属于代数又属于分析。目标是理解从估值的相互作用中产生的代数问题，以及围绕其定义所需的正则性性质的分析问题。 在分析方向，我们的主要目的是完善最近的结果Pokorny和Rataj的规则集定义的差异凸函数。在代数方面，在过去的十年里，我们看到了很大的进步，这主要是由S.在世纪早期的阿列斯克。从这个角度来看，经典的积分几何的Blaschke，Santalo，陈，等人，处理真实的空间形式在其完全等距群的作用下，似乎是一个更丰富的理论的平凡的基础情况。第一个非平凡的情况是埃尔米特积分几何，即复空间形式的积分几何。我们最近的结果表明，与这个空间家族相关的代数非常丰富，其特征在于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的新发现的能力，使计算在这些情况下，在一定程度上容易探索如何估值行为在黎曼流形没有这样广泛的对称性。

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.