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.
The subject of integral geometry concerns holistic measurements of geometric objects and their interactions. It has major ramifications in geometric probability, which may be characterized as variations on the Buffon needle problem: to calculate the average number of times a needle tossed randomly on a wooden floor crosses between planks. Since the turn of the century the subject of integral geometry has experienced a major resurgence, due largely to previously unexpected connections to algebra. The integral geometry of such simple geometric spaces as the flat plane, the round sphere and the hyperbolic plane was thoroughly studied in the 20th century. More complicated spaces, however, proved resistant to the technical means available at that time. The main thrust of our project was to exploit the new algebraic tools that have become available to analyze the next tier of geometric complication, namely the so-called complex space forms. These are geometric shapes that arise naturally from the special algebraic properties of the complex numbers. Using the new methods we were able to analyze these cases in great detail and completeness. Although the concrete geometric problems susceptible to this new theory are significantly more abstruse than the classical Buffon problem, they remain highly meaningful and are quite accessible in the sense that they can be be easily explained and motivated to a wide scientific public. Furthermore these ideas give rise to a new and powerful language for describing and manipulating geometric objects. The project also included work on other basic geometric problems. Notable among these was a thorough study of the so-called ropelength problem, i.e. the study of the configurations assumed by a perfectly elastic rope of a given thickness when tied in a knot. We were able to characterize such configurations in terms of a system of balancing forces. This enabled us to describe in great detail one simple yet particularly enigmatic example, the "simple clasp", in which two elastic strands of equal thickness are looped through each other and attached respectively to the ceiling and to the floor. The answer is startlingly complicated, each strand consisting of some eleven sections each governed by a different schema of force balance. Another secondary theme was the theory of "Monge-Ampere functions", which may be described as a type of geometric surfaces that may become very bumpy and and irregular, but only in such a way that the total degree of irregularity remains finite once quantified in a certain natural way. In addition the project contributed in a significant way to the education of the rising generation of mathematical professionals, through the involvement of two PhD students and one postdoctoral associate. Since then the two students have completed their degrees and gone on to professional appointments at major universities.
