An introduction to d-manifolds and derived differential geometry

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory
of derived differential geometry", available at
http://people.maths.ox.ac.uk/~joyce/dmanifolds.html
We introduce a 2-category dMan of "d-manifolds", new geometric objects which
are 'derived' smooth manifolds, in the sense of the 'derived algebraic
geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived
manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of
manifolds Man embeds in dMan as a full subcategory. We also define 2-categories
dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners",
and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". For brevity,
this survey concentrates mostly on d-manifolds without boundary. A longer and
more detailed summary of the book is given in arXiv:1208.4948.
Much of differential geometry extends very nicely to d-manifolds and
d-orbifolds -- immersions, submersions, submanifolds, transverse fibre
products, orientations, etc. Compact oriented d-manifolds and d-orbifolds have
virtual classes.
There are truncation functors to d-manifolds and d-orbifolds from essentially
every geometric structures on moduli spaces used in enumerative invariant
problems in differential geometry or complex algebraic geometry, including
Fredholm sections of Banach vector bundles over Banach manifolds, the
"Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer,
Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect
obstruction theories in algebraic geometry. Thus, results in the literature
imply that many important classes of moduli spaces are d-manifolds or
d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic
geometry.
D-manifolds and d-orbifolds will have applications in symplectic geometry,
and elsewhere.