Effective Computations in Arithmetic Mirror Symmetry (13frg165)


(University of Alberta)

(Bates College)

(American Mathematical Society)


The Banff International Research Station will host the "Effective Computations in Arithmetic Mirror Symmetry" workshop from to .

In string theory, Calabi-Yau varieties describe the extra dimensions of the universe, beyond the three spatial and one time dimension that we move through every day. Mirror symmetry is a mathematical interpretation of a conjecture first formulated by physicists, which states that Calabi-Yau varieties should occur in pairs. The main objective of this Focussed Research Group is to study the number-theoretic implications of mirror symmetry. More explicitly, we will study how mirror symmetry is reflected in the structure of the congruent zeta function for mirror pairs. Although tantalizing, the arithmetic implications of mirror symmetry have only been explored in a few special cases, due to the computational challenges of studying the number-theoretic properties of higher-dimensional varieties.
New advances in computational number theory offer a framework for computing and understanding the congruent zeta function. We will investigate experimentally the arithmetic mirror symmetry for a broad class of hypersurfaces, including non-diagonal hypersurfaces in projective space and specific families of hypersurfaces in toric varieties. This approach provides a powerful tool for making predictions about the arithmetic properties of general Calabi-Yau varieties and offers a means to investigate the properties of the zeta function for singular hypersurfaces. This research group brings together the expertise of an internationally recognized group of algebraic geometers, mathematical physicists, and computational number theorists.

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).