This project will involve the study of combinatorial methods in geometric topology and geometric methods in enumerative combinatorics and algebra. Most of this work will be in the context of quantum groups, quantum topological invariants, and finite-type invariants of 3-manifolds. The conjecture that the Jones polynomial distinguishes the unknot is one example of a problem that motivates this work. Another example is the conjecture that no Vassiliev invariant distinguishes a knot from its reverse. This research involves the investigation of connections between topology, enumeration, and quantum mechanics. Topology is the study of how objects such as knots and surfaces are connected to themselves; although it was first established firmly as a field of mathematics a century ago, it has seen major advances in the past 30 years due to contributions from many areas of mathematics and physics. Enumeration is the art of counting things, such as how many way that there are to tile a square tray with dominos; it is related to the thermal properties of materials and other questions in physics. Quantum mechanics is a famous subject, but it is less well known that it has had an enormous influence in mathematics; it has partly redefined our conception of geometry and mathematical spaces. These three areas saw a confluence in the 1980's, culminating in the Fields medals awarded to Vaughan Jones, Vladimir Drinfeld, and Edward Witten. The hope is to extend the work of these and other mathematicians and physicists.
