Award: DMS 1209387, Principal Investigator: Karsten Grove
The principal investigator plans to continue his work on global problems in differential geometry and related areas. Special emphasis will be devoted to the pursuit of global Riemannian geometry via additional structures. This includes but is not limited to structures arising from the presence of symmetries, and to structures arising from taking external or internal limits. Our efforts concerning investigations of relations between curvature, symmetry and topology is guided by the so-called symmetry program initiated by the investigator and set forth by the aim: "Classify or describe the structure of manifolds with positive or nonnegative curvature and large isometry groups". This area has experienced significant advances in various directions during the past two decades with contributions from many people, resulting in a number of classification type theorems, the discovery of new structures and numerous examples in non-negative curvature and one in the illusive case of positive curvature. Most recently rigidity phenomena providing a link to Tits geometry of buildings has emerged. Here topology of buildings is introduced via the Hausdorff topology of so-called chamber systems. The main focus of the project will be to further develop this connection to buildings, but will also include investigations of structures arising from the collapse of manifolds under a lower curvature bound, and recent connections between comparison geometry and a new applied area concerned with the "processing of manifold-valued data". The latter also deals with averages via non-linear "center of mass" and taking internal limits.
The project deals with a vast and flexible extension of the classical rigid and maximally symmetric euclidean, spherical and hyperbolic geometries, as well as of the theory of surfaces. Finding and exhibiting relations between geometry and topology is at the heart of the subject. Here geometry refers to those properties of a space that are invariant under distance preserving transformations (called symmetries here), whereas topology refers to the more flexible properties of a space that are invariant under transformations such as stretching, bending and deforming. Curvature governs the local behavior of geodesics, i.e., of the "straight lines" in the space. By comparison, the angle sum of a geodesic triangle in a positively curved space is bigger than 180 degrees, which is the angle sum of a triangle in the flat Euclidean plane. This general type of geometry plays a vital role in much of mathematics, physics and more recently in applications to signal processing and more. Most of the proposed activity falls under the umbrella of the "symmetry program" designed by the investigator. The main specific goals within the next few years are on the one hand to find additional new examples of positively and nonnegatively curved manifolds, and at the opposite extreme to show that the so-called symmetric spaces indeed are rigid objects in a natural sense. The latter is based on a recently discovered link to a very different area, namely Tits geometry of buildings, an area that has had profound and diverse applications within mathematics. Other important goals include developing new directions in geometry motivated by the emerging field of processing of manifold-valued data to, e.g., computer vision, medical imagining, sensor networks, and statistical analysis of shapes.
