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Mirror symmetry is a phenomenal relationship between symplectic and complex geometry.  First conceived by physicists, this still little-understood relationship, arose from the observation that two models of string theory - one related to Gromov-Witten invariants and the other to complex structure deformations - could lead to identical physical theories when respectively placed over two, topologically different, manifolds (used to model the "hidden" dimensions of our universe).  Homological mirror symmetry is the modern (conjectured) mathematical framework for this idea.  It proposes a categorical equivalence between the (derived category of) coherent sheaves on a complex manifold and the (derived) Fukaya category - whose objects are Lagrangian submanifolds (+ structure) - built on the symplectic structure of a different manifold called its mirror dual. > The mix of differential geometry, algebraic geometry, and homological algebra required to study this topic is generally foreboding, but in the case of Elliptic curves, both categories simplify enough to allow interested geometry enthusiasts a peak at this surprisingly strange relationship - which in this case will send lines of slope p/q in a symplectic 2-torus to rank p degree q vector bundles over a complex 2-torus. > In this talk I'll discuss the general idea of homological mirror symmetry and outline its details in the case of an elliptic curve.