A brief introduction to derived differential geometry

When two submanifolds meet non-transversely, their intersection may fail to be a manifold. However, such bad intersections can be dealt with as smooth objects in the setting of derived geometry, at the price of introducing higher categorical structure. Derived manifolds are objects constructed by iteratively taking fibered products, starting with smooth manifolds. We will show how such a theory can be derived from basic principles and explain its connection to derived algebraic geometry. We will then present a recent result that shows how the higher category of derived manifolds can be completely encoded by differential graded manifolds.