C^infinity Algebraic Geometry (with corners)

Manifolds, the main objects of study in Differential Geometry, do not have nice categorical properties. For example, the category of manifolds with smooth maps does not contain all fibre products.
The algebraic counterparts to this (varieties and schemes) do have nice categorical properties. 

A method to ‘fix’ these categorical issues is to consider C^infinity schemes, which generalise the category of manifolds using algebraic geometry techniques. I will explain these concepts, and how to translate to manifolds with corners, which is joint work with my supervisor Professor Dominic Joyce.