This project investigates the existence of polyhedral and Riemannian metrics of non-positive curvature on manifolds, and uses such metrics to study the topology of these manifolds. We are investigating how a combinatorial description of a 3-manifold gives rise to natural singular geometric structures, and to the existence of immersed surfaces. We also study the question of when non-positively curved polyhedral metrics can be deformed to smooth hyperbolic metrics. This project naturally leads to new approaches both to Thurston's Geometrization Conjecture and to the question of when homotopy equivalent manifolds are homeomorphic. We have found a number of examples of manifolds covered by bundles using polyhedral metrics and new conditions for finding immersed incompressible surfaces based on polyhedral curvature ideas.
Topological methods are particularly productive in low dimensional algebraic geometry.
Of particular interest in current research is the study of restrictions that a plane curve implies for the other fibres of its defining polynomial. These are of great interest in their own right, as well as in physics and the mathematical study of low-dimensional spaces. Important equations in the study of integrable systems can be understood in terms of the combinatorics of cubical and simplicial structures. These in turn have manifestations in higher-dimensional category theory, in number theory, and in the basic structure of combinatorial manifolds. The minimal Steiner tree problem asks for the shortest network of line segments that will interconnect a set of given points.