We will give an overview of the Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry in the setting of non-compact Calabi-Yau varieties given by the complement U of an anticanonical divisor D in a projective variety X. Namely, U is expected to carry a Lagrangian torus fibration, and a mirror Calabi-Yau variety U' can then be

constructed as a (suitably corrected) moduli space of Lagrangian torus fibers equipped with local systems. (Partial) compactifications of U deform the symplectic geometry of these Lagrangian tori by introducing holomorphic discs; counting these discs yields distinguished regular functions on the mirror U'. The goal of the talk will be to illustrate these concepts on simple examples, such as the complement of a conic in C^2.

If time permits we will also try to explain the relation of this story to the symplectic cohomology of U and its product structure

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