Quantum Curves and Quantum Knot Invariants

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).

BIRS Scientific Director, Nassif Ghoussoub
E-mail: birs-director[@]birs.ca
http://www.birs.ca/~nassif