# Mirror Symmetry

## Content

Mirror Symmetry investigates a strange connection between two types of geometry: symplectic geometry and algebraic geometry. It was originally discovered in theoretical physics as a duality between two models of string theory: the A- and the B-model. In the 1990's it also became important in mathematics because it could be used to calculate numbers in geometry that mathematicians had tried to find for many years. Since then it has become a main research topic in geometry, algebra and mathematical physics.

In this course we will explore the basic ideas behind mirror symmetry from the point of view of the homological mirror symmetry conjecture. This conjecture formulates an equivalence between two categories: the Fukaya category of a symplectic manifold and the derived category of coherent sheaves of an algebraic variety. We will introduce the mathematics needed to define these two categories such as homology, A-infinity algebras, Floer theory and derived categories. These concepts will be illustrated by some basic examples coming from surfaces.

## Topics

- Motivation from physics
- Homology and Cohomology
- The A-infinity Formalism
- A first glance at Mirror Symmetry
- A-models
- B-models
- Examples of Mirror symmetry