We discuss four generalizations and applications of $C^\infty$-schemes with corners.

1. 1. To provide a model for a theory of ‘Synthetic Differential Geometry with corners’.

2. 2. To a theory of ‘ $C^\infty$-stacks with corners’. $C^\infty$-Stacks are studied in D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. Most of the theory extends to corners with no changes.

3. 3. To a theory of ‘ $C^\infty$-schemes with a-corners’. ‘Manifolds with a-corners’ are introduced in D. Joyce, arXiv:1605.05913 as a class of manifolds with corners with an alternative smooth structure, that has applications in analysis, e.g. Morse theory moduli spaces should really be manifolds with a-corners, not corners.

4. 4. To a theory of ‘derived $C^\infty$-schemes and derived $C^\infty$-stacks with corners’, and within these, ‘derived manifolds with corners’ and ‘derived orbifolds with corners’, where ‘derived’ is in the sense of Derived Algebraic Geometry.

This chapter summarizes D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. If X is a manifold then the set $C^\infty$(X) of smooth functions on X is a ‘ $C^\infty$-ring’, a rich algebraic structure with many operations. A module over a $C^\infty$-ring is a module over it as an real algebra. A $C^\infty$-ring has a cotangent module. For $C^\infty$(X) this is the sections of the cotangent bundle T*X.

‘ $C^\infty$-Schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry, and of allowing Differential Geometers to study spaces far more general than manifolds. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to ‘derived manifolds’ in Derived Differential Geometry. The category of $C^\infty$-schemes has good properties: it is Cartesian closed and has all finite limits and directed colimits.

We also study sheaves of modules over $C^\infty$-schemes, as for (quasi-)coherent sheaves in Algebraic Geometry. A $C^\infty$-scheme has a cotangent sheaf, generalizing the cotangent bundle of a manifold.