Hyperkaehler Geometry with Applications

What is common in (1) the existence of a magnet with a single pole (2) the reliability of computer networks and (3) code theory and code breaking? The proposed research provides an answer: these scientific problems can all be attacked using quaternionic geometry. Quaternions are four dimensional analogues of complex numbers. For problem (1) one can study magnetic monopoles using quaternionic equations. The possible existence of these and similar elementary particles could lead to new energy sources. For (2) the proposed research shows that the number of holes on a certain quaternionic surface attached to a graph agrees with the reliability polynomial of a computer network based on the graph. Qualitative properties of this reliability polynomial, obtained from the study of the geometry of quaternionic surfaces, help explain how to make computer networks, like the internet, more reliable. In (3) arithmetic study of certain quaternionic surfaces sheds light on the representation theory of finite groups of Lie type, which are used in various schemes in code theory. Information emerging from the geometry of these quaternionic surfaces, could help devise better codes. In short, the proposed research is two-folded, first it studies fundamental problems in quaternionic geometry, and second it breaths life into these investigations by applying the results to other fields in mathematics and physics. This yields a colourful palette of various fields in mathematics and physics all related in one way or another to quaternionic geometry.This proposal therefore aims to understand the global analysis, geometry, topology and arithmetic of complete hyperkaehler manifolds of non-compact type and find exciting applications in other fields of mathematics and physics, where these manifolds naturally appear. The proposed research has two main aspects: studying fundamental questions for non-compact hyperkaehler manifolds, such as Hodge theory and the Atiyah-Singer index theorem, and applying these methods in other fields. The hyperkaehler spaces appearing in this proposal include: moduli spaces of Yang-Mills instantons on asymptotically locally Euclidean gravitational instantons; more generally Nakajima's quiver varieties; toric hyperkaehler varieties; moduli spaces of magnetic monopoles on R^3; moduli spaces of Higgs bundles on a Riemann surface; and more generally hyperkaehler spaces appearing in the non-Abelian Hodge theory of a curve (like moduli of flat GL(n,C)-connections and character varieties) and in the Geometric Langlands Program. The fields of applications include: combinatorics, representation theory, finite group theory, low dimensional topology, number theory, mathematical physics and string theory.