Thang Le plans to continue his study of quantum and finite type invariants of links and 3-manifolds. In particular, he would like to study problems arising around the volume conjecture which connects quantum invariants to classical objects like fundamental groups, torsions, and volumes. Other problems involve integrality properties (in broad sense) of quantum invariants and the topology behind them, and their applications. The field has interactions with geometry, combinatorics, number theory, and physics.
The theory of knots and 3-manifolds is an old branch of mathematics which has gained renewed interest among mathematicians and physicists after the discovery of the Jones polynomial and its relation to theoretical physics (quantum field theory, high energy physics). In fact, it is now one of the most active domains in mathematics. Many results of knot theory may also find applications in molecular biology. To classify knots and 3-manifolds, mathematicians use "invariants". This research project studies new classes of invariants of knots and 3-manifolds and their relationships with the classical ones. The new invariants are very powerful in distinguishing knots and 3-manifolds.
