Geometric and quantum invariants of knots -- Ilya Kofman, February 1, 2005

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The works of Thurston and Jones revolutionized low-dimensional topology. 
 Thurston established the importance of geometric invariants, especially 
hyperbolic volume, for 3-manifolds.  The discovery 20 years ago of the Jones 
polynomial led to vast families of quantum invariants, obtained from diagrams 
of knots and related objects.  Understanding quantum knot invariants in terms 
of the geometry of the knot complement remains a deep open problem.  I will 
describe the surrounding mathematics and discuss some recent progress, 
focusing on the Jones polynomial and hyperbolic volume.