Calabi-Yau geometry and mirror transforms of the Hodge conjecture

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Since the mid 1990's mathematicians have proved stronger and more complete "mirror theorems" relating the complex geometry to the symplectic geometry of Calabi-Yau manifold pairs. Mirror symmetry permits a problem in complex geometry on one manifold to be replaced by equivalent, reformulated problems in symplectic geometry on its mirror. In this way intractable questions on one side may admit a simple answer on the other. The Hodge Conjecture has been described as "a metaphor for transforming transcendental computations into algebraic ones." The most important conjecture in complex algebraic geometry is much more than this, and yet it is merely one of a host of related conjectures involving algebraic cycles. We have recently obtained striking results relating microscopic black holes and special values of L-functions, quantities at the heart of some of the deepest conjectures in number theory. This work establishes the sensitivity of the mirror transform to the arithmetic structure of algebraic cycles. Building on this foundation, and also related work on modular forms, K3 surfaces, and toric geometry, we will apply mirror symmetry to tackle several variants of the Hodge Conjecture. By looking at state-of-the-art results on Calabi-Yau geometry "through the looking glass", we can establish consequences of the Hodge Conjecture in entirely new ways.