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.
was held from March 20-25, 2011 in Les Diablerets, Switzerland at La maison des congres and was supported under DMS-1106739. The goal of the conference was to bring together world-class experts and young researchers in the field to discuss recent advances in the topics of Quantum Invariants, Hyperbolic Geometry, Teichmueller Space, Volume and AJ Conjectures and Chern-Simons Theory. NSF funding was used to support travel expenses for 7 researchers and 4 graduate students from US institutions and one researcher from the United Kingdom. Twelve countries were represented by the 76 attending participants. Participant selection emphasized young researchers with a promising research record, minorities when available, graduate students and international researchers with no other support. Intellectual Merit: The main idea of the conference was to have a number of mini-courses followed by working seminar talks. To encourage informal discussions and exchange of ideas, all the speakers' accommodations and meals were in the same place (Hotel les Sources). Broader Impact: The conference brought together leading experts in 3-dimensional Geometry, Topology, and Quantum Field Theory, and provided a great opportunity to attract and educate young researchers and curious minds. There were 17 50-minute presentations given by invited speakers.