Graded manifolds, Lie algebroids, and equivariant cohomology. Part 1

The goal of this minicourse is to explain how equivariant cohomology can be formulated in terms of differential graded manifolds and Lie algebroids. In this language, the constructions of the Weil and the BRST model, as well as the Mathai-Quillen isomorphism between them are concise, natural, and relatively straightforward. In Part 1 I start with a self-contained introduction to the basic notions of graded differential geometry, where the grading is over an arbitrary abelian group, typically $\mathbb{Z}$ or $\mathbb{Z}^2$.