Spring School in Geometry and Quantum Topology

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Modern Knot Theory started with the seminal discovery of the Jones polynomial by V. Jones in the mid eighties. Shortly after Jones's discovery, a plethora of polynomial invariants were constructed using the theory of Quantum Groups, and the later developed TQFT. Among the giants in the field are Fields Medalists V. Drienfeld, M. Kontsevich, V. Jones and E. Witten. Although the Jones polynomial is a combinatorially defined object, certain limits of it contain important information about the algebraic topology and the differential geometry of the knot complement. Three such limits are the focus of the school: namely the Volume Conjecture, the AJ Conjecture and Hyperbolic Geometry. In addition, the asymptotics of the Jones polynomial is an important example of a solvable topological quantum field theory, which illustrates some of the latest ideas in string theory and M-theory. The geometric limits also contain nontrivial new arithmetic of 3-dimensional objects, which is also one of the themes of the school.