Gauge theory and low-dimensional topology

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Manifolds are geometric shapes that, near any point, look like Euclidean space even though their global structure may be twisted and curved in interesting ways. The surface of a ball provides a concrete example: even though it is not flat, one can repair any small puncture with a little rectangular patch. Since manifolds are all locally indistinguishable, mathematicians search for invariants that reflect the manifold's global curving and twisting. For example, imagine a near-sighted insect living on either the surface of a ball or a flat sheet of paper. How could it tell the two apart? One method is to compute an invariant called the Euler characteristic, which is simply the number V - E + F (the number of vertices minus the number of edges plus the number of faces in a triangulation). This number is actually independent of the triangulation and hence an invariant.     One of the most important problems in low-dimensional topology is that of distinguishing all 3-dimensional manifolds. Understanding the global structure of these manifolds has numerous applications to physics, chemistry and biology. The applicant proposes new invariants of 3-manifolds and new methods for computing them. The long-term goal is to use the invariants to help in classifying 3-manifolds and knots and links contained in them.