A fundamental tenet of arithmetic geometry is the notion that one can infer arithmetic properties of a variety from its geometry. The investigator's research primarily deals with such a notion. Specifically, the PI hopes to show that the number of rational points with bounded Weil height in an orbit under the action of the group of automorphisms of a K3 surface grows asymptotically like a power of the bound. The exponent of growth should depend only on the geometry of the surface. When the K3 surface contains no -2 curves, the exponent is expected to be n-2, where n is the dimension of the Picard group. The situation is particularly interesting when there are -2 curves. In this case, the exponent of growth should be related to the Kahler cone or nef cone. The Kahler cone lies in a Lorentz space of dimension n (where the intersection product is the Lorentz product) and a cross section of the cone lies in a hyperbolic space of dimension n-1. When the K3 surface has a -2 curve, the boundary at infinity of this cross section is a fractal on a sphere of dimension n-2. The PI believes that the previously mentioned exponent of growth should be the Hausdorff dimension of this cross section. When n=4, these fractals are similar to the well known Apollonian gasket.
This is basic research aimed at improving our understanding of K3 surfaces and related objects. Surfaces of type K3 can be thought of as higher dimensional analogs of elliptic curves, for which analogous research was the subject of early 20th century mathematics. Our basic understanding of elliptic curves eventually led to the development of elliptic curve based public key codes, fast factoring algorithms, and efficient error correcting codes. The theory and existence of public key codes revolutionized cryptography and with it, the internet as we know it. Without them, internet based commerce would be far too vulnerable to allow the sorts of transactions we currently take for granted. A K3 surface can also be thought of as a two dimensional Calabi-Yau manifold. Three dimensional Calabi-Yau manifolds are of interest to physicists studying string theory and quantum field theory.
