The Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving **Homological Mirro****r Symmetry (HMS)** in full generality, and fully exploring its applications.

**Mirror symmetry**, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model).

We envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:

- categorical algebraic geometry and
**non-commutative (nc) spaces**: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry. - the
**Strominger-Yau-Zaslow (SYZ)**approach, which provides a global geometric prescription for the construction of mirror pairs. - Lagrangian Floer theory and
**family Floer cohomology**, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.