Mathematical Sciences: Foundations of Integral Geometry

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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.