Invariants of Links and 3-manifolds
Thang T. Q. Le Principal Investigator
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DMS-0204158<br/>Thang Le<br/><br/>Thang Le plans to continue his study of quantum and finite type<br/>invariants of links and 3-manifolds. In particular, he would like<br/>to study problems arising around the volume conjecture which<br/>connects quantum invariants to classical objects like fundamental<br/>groups, torsions, and volumes. Other problems involve integrality<br/>properties (in broad sense) of quantum invariants and the<br/>topology behind them, and their applications. The field has<br/>interactions with geometry, combinatorics, number theory, and<br/>physics.<br/><br/>The theory of knots and 3-manifolds is an old branch of<br/>mathematics which has gained renewed interest among<br/>mathematicians and physicists after the discovery of the Jones<br/>polynomial and its relation to theoretical physics (quantum field<br/>theory, high energy physics). In fact, it is now one of the most<br/>active domains in mathematics. Many results of knot theory may<br/>also find applications in molecular biology. To classify knots<br/>and 3-manifolds, mathematicians use "invariants". This research<br/>project studies new classes of invariants of knots and<br/>3-manifolds and their relationships with the classical ones. The<br/>new invariants are very powerful in distinguishing knots and<br/>3-manifolds.