Research in Modern Integral Geometry

点击翻译按钮获取中文摘要

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.