Award: DMS 1406252, Principal Investigator: Joseph H. G. Fu
The subject of integral geometry grows out of the need to calculate the statistics of random shapes or objects. The basic idea is to begin by isolating certain fundamental quantitative geometric measurements that are finitely additive, in the sense that if an object is formed by amalgamating two smaller objects then the quantity associated to the total is equal to the sum of the quantities associated to the constituents, minus the quantity associated to their overlap. Such quantities, called valuations, include the volume and the surface area of a three-dimensional object, or the area and the perimeter of a two-dimensional figure. The statistics of how a given valuation varies over families of intersecting objects turns out to be governed by a rich system of algebraic rules, which constitute the focus of this project. Integral geometry should thus be viewed as a fundamentally important aspect of geometry as a whole, with many possible applications to the sciences.
The methods involved belong both to algebra and analysis. The objectives are to understand the algebraic questions arising from the interactions of valuations, and also the analytic problems surrounding the regularity properties necessary for their definition. In the analytic direction our main aim is to refine recent results of Pokorny and Rataj on the regularity of sets definable by differences of convex functions. On the algebraic side, the past decade has seen a great deal of progress, largely catalyzed by the fundamental new algebraic structures on valuations introduced by S. Alesker in the early years of the century. From this perspective the classical integral geometry of Blaschke, Santalo, Chern, et al., dealing with the real space forms under the actions of their full isometry groups, appears as the trivial ground case of a much richer theory. The first nontrivial case is that of Hermitian integral geometry, i.e. the integral geometry of the complex space forms. Our recent results reveal that the algebra associated with this family of spaces is extremely rich, characterized by many mysterious and unexplained phenomena. One of our primary goals is to exploit our new-found ability to make computations in these cases with some degree of ease to explore how valuations behave in Riemannian manifolds without such extensive symmetries.
