Gauge theory and low dimensional topology

点击翻译按钮获取中文摘要

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 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 would be to compute the Euler characteristic V E + F, the number of vertices minus the number of edges plus the number of faces in a triangulation, which is independent of the triangulation and is an example of a topological 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 knots inside them, and he proposes new methods for computing these invariants. The invariants are defined using gauge theoretic methods applied to character varieties, and there are a number of interesting student projects that are an important part of the research program. The long-term benefits of this research program are two-fold: the knowledge gained will help determine to what extent gauge theory can deliver new invariants that can be used to help classify 3-manifolds and knots, and the training program will produce highly qualified personnel at the undergraduate, postgraduate, and postdoctoral levels with research skills in geometric topology.