Towards Homological Mirror Symmetry

Abstract

This is an expository article on the A-side of Kontsevich's Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_\infty$-categories and their homological algebra, and later restrict to Fukaya categories, with particular emphasis on the basics of the underlying Floer theory, and the geometric features therein. Finally, we place the theory in the context of mirror symmetry, building towards its main predictions.