Mathematical Sciences: Foundations of Integral Geometry

主要研究者将找到一类自然的物体，其曲率测量是定义良好的，并且满足运动学公式。期望这样的一类将由满足某些附加假设的非负“蒙日-安培”函数的零集组成。他将研究曲率测度是否是内禀不变量。亚历山德罗夫关于高维曲面内禀曲率的理论将被推广。当标量曲率被认为是度量时，非光滑物体的几何可以从一个统一的角度来研究。没有一个综合的曲率测度理论既包含凸集又包含奇异代数变体。首席研究员将使用他最近开发的涉及几何测量理论的方法来进一步澄清这个问题。使用这些技术的新工具将被创造出来。

The principal investigator will find a natural class of objects on which curvature measures are well-defined and satisfy the kinematic formula. It is expected that such a class will consist of the zero-sets of non-negative "Monge-Ampere" functions satisfying certain additional assumptions. He will investigate whether curvature measures are intrinsic invariants. Alexandrov's theory of intrinsic curvature of surfaces to higher dimensions will be generalized. The geometry of non-smooth objects can be studied from a unified point of view when scalar-valued curvatures are thought of as measures. No comprehensive theory of curvature measures embraces both convex sets and singular algebraic varieties. The principal investigator will use his recently developed methods involving geometric measure theory to further clarify this problem. New tools using these techniques will be created.