The overall theme of this project is to explore the deep connections between convex geometry and combinatorial lattice theory, and to pursue applications in each direction. From the geometric viewpoint, this project will focus particular attention on the structure of convex bodies, star-shaped sets, mixed volumes, and dual mixed volumes, with a goal of characterizing valuations and set functions that are invariant under various group actions and applying these results to problems in geometric probability, convex geometry, geometric tomography, and analysis on Grassmannians. This project will also pursue an investigation of the many deep and previously unexploited connections between convex geometry and algebraic combinatorial theory, with the particular end of the development of a combinatorial theory of invariant valuations and kinematic formulas on finite lattices, (where the invariance is with respect to the action of an automorphism group). A central goal of this investigation is the development and application of combinatorial analogues to Hadwiger's characterization theorem for invariant valuations and to classical kinematic formulas in the context of partially ordered sets, with a special focus on the combinatorial structures that arise in convex geometry. Convex geometry and the theory of valuations treat the fundamental question of how to measure (or in the case of finite features, to enumerate) and ultimately to characterize intrinsic features of geometric objects. Examples include the reconstruction of information about a geometric object from limited data, such as information about projections and shadows (stereology) or slices and cross-sections (tomography). These techniques lead in turn to many applications, such as those in biotechnology (such as molecular biology), economics and finance (analysis of efficient and equitable distributions of limited resources over a population), and computer graphics (the visual display of information).
