Invariants of Links and 3-manifolds

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DMS-0204158Thang LeThang Le plans to continue his study of quantum and finite typeinvariants of links and 3-manifolds. In particular, he would liketo study problems arising around the volume conjecture whichconnects quantum invariants to classical objects like fundamentalgroups, torsions, and volumes. Other problems involve integralityproperties (in broad sense) of quantum invariants and thetopology behind them, and their applications. The field hasinteractions with geometry, combinatorics, number theory, andphysics.The theory of knots and 3-manifolds is an old branch ofmathematics which has gained renewed interest amongmathematicians and physicists after the discovery of the Jonespolynomial and its relation to theoretical physics (quantum fieldtheory, high energy physics). In fact, it is now one of the mostactive domains in mathematics. Many results of knot theory mayalso find applications in molecular biology. To classify knotsand 3-manifolds, mathematicians use "invariants". This researchproject studies new classes of invariants of knots and3-manifolds and their relationships with the classical ones. Thenew invariants are very powerful in distinguishing knots and3-manifolds.