This project involves the fields of sphere packing and arithmetic geometry of K3 surfaces. Kumar will continue his research on sphere packing in high dimensions and related problems. The central question in sphere packing asks for the densest packing of Euclidean space by congruent non-overlapping spheres. The answer is known only in dimensions up to 3, and in general this is a very difficult though natural question. Cohn and Kumar have made progress in 8 and 24 dimensions, showing that the sphere packings coming from the E_8 and Leech lattices are very close to being the densest packings in these dimensions. They are investigating related problems, such as that of sphere packing on the surface of a sphere (spherical codes), hyperbolic sphere packing, and how to make these using potential energy minimization. Kumar will also work on the arithmetic of K3 surfaces: projective algebraic smooth surfaces with trivial canonical bundle and zero irregularity. Their geometry and moduli have been studied exhaustively by algebraic geometers, but questions about their arithmetic, such as rational points, connections with modular forms, etc., are not so well understood. Kumar has investigated the connection between some families of K3 surfaces and curves of genus two, with arithmetic applications such as finding elliptic curves of high rank and computing equations of Hilbert modular surfaces.
This project will focus on the areas of sphere packing and arithmetic geometry. The sphere packing question asks for a packing of space by equal sized non-overlapping spheres, so as to cover the maximum possible fraction of volume. Stated as a problem in geometry, it nevertheless has many connections to other parts of mathematics, such as number theory, group theory and Lie theory. It also has many important applications outside of mathematics, for instance, in coding and communication theory, and in physics. For instance, it is useful in connection with error correcting codes used on CDs, spherical codes used in cellphone signals, and radio or more general analog signals. Many related problems, such as minimization of potential energy, are of interest to physicists and materials scientists, for instance, in the self-assembly of nanoparticles. The PI, Kumar, and his collaborators have made progress in understanding sphere packing and related problems in high dimensions, and this project will continue that research. The field of arithmetic geometry applies techniques from algebra and algebraic geometry, which is the study of the geometry of sets of solutions to polynomial equations, to the field of number theory. For instance, one might wish to know the solutions in integers or rational number to a certain system of polynomial equations (such as Fermat's equation, x^n + y^n = z^n). Kumar's research focuses on the study of the arithmetic of K3 surfaces. These beautiful objects have connections to many parts of mathematics and physics, such as lattice theory and mirror symmetry. K3 surfaces and related geometric objects are of fundamental interest in string theory, which tries to understand the physical nature of the universe. This project will further the research into the number-theoretic properties of such surfaces.
