Quantum Curves and Quantum Knot Invariants (14w5073)


Vincent Bouchard (University of Alberta)

(Columbia University)

(University of California, Davis)

Alexei Oblomkov (University of Massachusetts)

Marko Stošić (Instituto Superior Técnico, Portugal)

(University of Warsaw)


The Banff International Research Station will host the "Quantum Curves and Quantum Knot Invariants" workshop from June 15th to June 20th, 2014.

Mirror symmetry among Calabi-Yau spaces is an idea originated in
modern theoretical physics. After the mathematical abstraction of
Kontsevich, mirror symmetry can be formulated without referring to
spaces. For example, we can ask what is the mirror partner of Catalan
numbers. The answer is the generating function of the Poincaré
polynomials of the space of all Riemann surfaces. From this point
of view we obtain a Schrödinger equation that determines this
generating function as a solution.

Similarly, we can ask for a mirror symmetry description of the
Jones polynomials of knots. In 2011-2012 a conjectural formula
was discovered through many works of physicists and mathematicians.
The corresponding Schrödinger equation turns out to be the same as
the conjectural equation proposed by a mathematician a decade ago.
This mirror symmetric idea has led physicists to extend the formulas
toward more refined knot invariants such as Khovanov homology.

There is strong evidence that these conjectures are true, based on
numerous computer checks and rigorous mathematical results of special
cases. Since the subject area has connections to many different fields
of mathematics and theoretical physics, this workshop is designed to
bring experts together for the purpose of better understanding the
exciting new developments. Reflecting that many key players are young
scholars, the workshop is expected to attract a young generation of
researchers with diverse background.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).