Geometry and Physics of Quantum Curves (18w5078)


(Central Michigan University)

(University of Pennsylvania)

Marco Gualtieri (University of Toronto)

(University of California, Davis)


The Banff International Research Station will host the "Geometry and Physics of Quantum Curves" workshop from September 9th to September 14th, 2018.

At the time of its birth, quantum mechanics was a mystery. People asked: ``Yes, it explains experiments. But why does it work?" From the 21st century perspective, quantum theory looks to have popped up instantaneously. In reality, it took over 30 years to be developed. Only very slowly mathematicians and physicists came to the understanding of quantum theory in the early 20th century.

100 years later, we are encountering a similar excitement and mysteries in geometry. On one side, we have a number of data coming from symplectic geometry known as Gromov-Witten invariants. The mirror symmetry idea tells us that they should also come from holomorphic geometry of complex manifolds. So far we have not identified the full scope of the holomorphic geometry that explains quantum invariants.

Right at this moment, there are three amazing developments in this area of research. One is the analysis of Fredholm determinants of quantum operators that mysteriously reproduces the quantum invariants. The second is holomorphic geometry of the very notion of quantization. And the third is a beautiful theory of quantizing Hitchin spectral curves into differential operators. The proposed workshop aims at identifying these ideas behind the quantization, and to produce a more perfect union of disparate theories.

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).