The mirror conjecture predicts a remarkable equivalence, discovered first by conformal field theorists, between algebraic geometry of holomorphic curves in a Calabi-Yau manifold and properties of Picard-Fuchs differential equations describing variations of complex structures on the mirror partner of this manifold. Precise mathematical formulations of the conjecture have been worked out for Calaby-Yau complete intersections in toric varieties and tie together combinatorics of dual polyhedra, higher hypergeometric functions, their differential equations and integral representations.
In the talk, we will show how the mirror phenomenon can be understood in terms of homological geometry: one should not hesitate to solve algebraic and differential equations, compute residues and integrate differential forms, or even do Morse theory in the manifolds whose coordinate rings are cohomology algebras of some topological spaces.