Volume 8 (2006)

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[Group photograph]

In the summer of 2001, we (Dave and Jim) were at the Gokova conference in Turkey talking about BIRS, the new mathematics institute that was going to open in Banff, Canada in 2003. Although 2003 seemed like a long way off at the time, we wanted to propose a workshop. Fortuitously, Jim had recently heard about some exciting work in physics by Gopakumar, Vafa and others that had found some very explicit connections between topological string theory and Chern–Simons gauge theory – the very same physical theories that led to the mathematical theories of Gromov–Witten invariants and finite type invariants. Although these ideas had not yet taken hold in the math community, it seemed likely that within a few years they would be timely and warrant a workshop.

Indeed, by 2003, the topic was very timely. Physicists Aganagic, Klemm, Marino, and Vafa had developed the "topological vertex", a gadget which (conjecturally) computed Gromov–Witten of toric Calabi-Yau threefolds in terms of certain invariants of finite type: Chern–Simons invariants. Mathematicians Li, Liu, Liu, and Zhou had begun to develop a mathematical framework for the topological vertex. Garoufalidis and Le had just proven the LMOV conjecture. This conjecture encoded integrality properties of the HOMFLY(PT) polynomial that must hold if the conjectural large N duality was indeed true. In addition new conjectures relating large N duality with Khovanov homology were forming.

Early in 2003, almost two years after having conceived of the workshop, the organizers conceived (independently!) of something else: children. Jenny (Jim's wife) and Andrea (Dave's wife) were both expecting and coincidentally had the same due date: November 1st, only two weeks before the workshop! On October 31st, while the organizers were busy trying to prepare the workshop's program, the first little disorganizer, John Robert Auckly, was born. The second little disorganizer, Reed Frazier Bryan, decided to wait until November 15th, the first day of the workshop to be born. Despite the valiant attempts of our beloved disorganizers to disrupt the event, the workshop came and went without a hitch. Jim stayed at home and Dave, two weeks into fatherhood, went to BIRS to oversee the event.

It is easy to organize a successful program at the Banff Research Station. The surroundings are inspirational. Our Physics coach (Marcos Marino) gave an equally inspiring series of talks on large N duality. Additional lecture series were given by M Liu, J Li, D. Bar-Natan and H Wenzl. The other speakers were: S Garoufalidis, M Hutchings, T Kimura, C Leung, L Ng, J Przytycki, J Roberts, J Sawon, and J Shapiro.

These proceedings reflect the wide array of topics related to Gromov–Witten invariants, finite type invariants, and the recently discovered connection between them. The first three papers are reprints of three of the first mathematical papers related to large N duality and Chern–Simons invariants. We have included papers of both an expository and research nature submitted by participants. The final paper is an expanded set of lecture notes from a portion of the Marino lectures. We hope these proceedings motivate others to investigate this fascinating area of mathematical research.

David Auckly and Jim Bryan, June 2005