Research in Modern Integral Geometry

该项目旨在建立在积分几何的最新决定性进展的基础上，该进展主要源于S.Alesker在凸赋值方面的工作。Alesker已经证明，估值空间受制于一系列自然的代数运算，包括可交换乘法。此外，对凸集的限制是不自然的，Alesker在一般光滑流形M上引入了赋值理论，用M的某一类奇异子空间代替凸体。从这个角度来看，布拉施克、陈、费德勒等人的经典作品。可以看作是平凡的基本情况，对应于全旋转群So(N)，这是一个更一般的理论，适用于在球面上传递作用的较小的群。该项目将在几个不同的背景下探索这些想法，特别是(真实的)双曲空间和复杂的空间形式。在一个不同的方向上，赋值理论开辟了一种新的、看似非常自然的方法来研究Finsler流形的几何。该项目的另一个方面解决了哪些奇异子空间X对于赋值理论来说是真正自然的问题。PI的早期工作给出了一个形式上的答案-X必须承认‘’正常循环‘-但这个条件本身很难理解。最后，在最后几个问题中出现的分析风格似乎与’‘长度问题’‘有一个无限维的类比。到目前为止，在这个方向上的工作只涉及一阶理论，而更有成果的二阶方面还没有在这个无限维世界中出现。庞加莱的经典公式说，如果两条曲线随机放置在一个球面上，那么平均而言，交点的数量与它们的长度的乘积成正比。这个公式只是根据所涉及物体的几何特征表示自然概率测量的一系列类似公式中的一个。事实证明，这些测量值本身受制于一种代数，这种代数以神秘的方式与底层几何联系在一起。我们将在各种背景下探讨这些问题。

The project aims to build on recent decisive progress in integral geometry arising mainly from the work of S. Alesker on convex valuations. Alesker has shown that the space of valuations is subject to a a range of natural algebraic operations, including a commutative multiplication. Furthermore the restriction to the convex setting turns out to be unnatural, and Alesker has introduced a theory of valuations on general smooth manifolds M, with a certain class of singular subspaces of M replacing the convex bodies. From this perspective the classical work of Blaschke, Chern, Federer et al. may be viewed as the trivial ground case, corresponding to the full rotation group SO(n), of a more general theory applying to smaller groups that act transitively on the sphere. The project will pursue these ideas in a few different contexts, specifically (real) hyperbolic spaces and complex space forms. In a different direction, the theory of valuations opens up a new and seemingly very natural approach to the geometry of Finsler manifolds. Yet another aspect of the project addresses the question of which singular subspaces X are truly natural for the theory of valuations. A formal answer is provided by earlier work of the PI--- X must admit a ``normal cycle"--- but this condition is itself very poorly understood. Finally, the style of analysis arising in these last questions appears to have an infinite-dimensional analogue in the ``ropelength problem." Work to date in this direction has involved only the first order theory, while the more fruitful second order aspects have not yet been made their appearance in this infinite-dimensional world.The classical formula of Poincare states that if two curves are placed at random on a sphere then, on average, the number of points of intersection is proportional to the product of their lengths. This formula is just one of an array of similar formulas expressing natural probabilistic measurements in terms of the geometric characteristics of the objects involved. It turns out that these measurements are themselves subject to a kind of algebra, which is linked in mysterious ways to the underlying geometry. We will explore these issues in a variety of contexts.