





I am writing a book, which is under contract to be published by Oxford University Press, hopefully to appear by the end of 2018. If you are interested, you could start with the following survey paper:
D. Joyce, 'An introduction to dmanifolds and derived differential geometry', pages 230281 in L. BrambilaPaz, O. GarciaPrada, P. Newstead and R.P. Thomas, editors, 'Moduli spaces', London Mathematical Society Lecture Note Series 411, Cambridge University Press, 2014.Here too is a "detailed summary" of the book, a longer survey paper, which basically corresponds to Chapter 1 and Appendix A of the preliminary version of the book, and contains moreorless all the main definitions and theorems, but no proofs.
D. Joyce, 'Dmanifolds, dorbifolds and derived differential geometry: a detailed summary', arXiv:1208.4948, version 2, December 2012. 173 pages.
You can download a preliminary version of the book (from 2012) here:
'Dmanifolds and dorbifolds: a theory of derived differential geometry', pdf file. 768 pages.
Warning: the final OUP version of the book will use a different definition of dmanifolds to the three 2012 papers/books above, which is moreorless equivalent, but shorter and nicer to use; and it will have stronger theorems, will be better than the preliminary version in lots of other ways, and may even be shorter. I am not going to make the OUP version of the book available online, or any draft more recent than 2012, as this would make OUP cross; so if you want to read it when it is published, you had better ask your librarian to order a copy, or even better buy one yourself.You can download the slides of a talk I gave on it in London in December 2015 here:
'Derived differential geometry'. pdf file
In August 2015 I gave a 14 hour lecture course on 'Derived
differential geometry' to a summer school in Aarhus. You can download
the slides of the lectures from the course
web page.
Here is a survey paper on C^{∞}schemes and C^{∞}algebraic geometry which may be useful: arXiv:1104.4951.
Here is a brief description of what the dmanifolds book is about. See the reference list at the end for relevant literature.
The book describes and studies a new class of geometric objects I call "dmanifolds". Dmanifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The closest thing to them in the literature is the "derived manifolds" of David Spivak. But dmanifolds are rather simpler than Spivak's derived manifolds  dmanifolds form a 2category which is constructed using fairly basic techniques from algebraic geometry, but derived manifolds form an ∞category (simplicial category) which uses advanced ideas like homotopy sheaves and Bousfeld localization.
Following Spivak, my dmanifolds are based on the idea of "C^{∞}schemes". These were invented in Synthetic Differential Geometry (see Dubuc, 1981, and my 2010, 2011 papers below), and are a way of applying the methods of Algebraic Geometry in the world of real smooth geometry to get a large class of geometric objects which include smooth manifolds, but also many singular spaces.
Manifolds are examples of dmanifolds  that is, the category of manifolds embeds as a subcategory of the 2category of dmanifolds  but dmanifolds also include many spaces one would regard classically as singular or obstructed. A dmanifold has a virtual dimension, an integer, which may be negative. Almost all the main ideas of differential geometry have analogues for dmanifolds  submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on  but the derived versions are often stronger. For example, the intersection of two submanifolds in a manifold exists as a manifold if the intersection is transverse, but it always exists as a dmanifold. There are also good notions of dmanifolds with boundary and dmanifolds with corners, and orbifold versions of all this, dorbifolds. I claim that the (morally and aesthetically) "right" definition of Kuranishi space in the work of FukayaOhOhtaOno is that they are dorbifolds with corners.
A useful property of dmanifolds and dorbifolds is that they have wellbehaved virtual cycles or virtual chains. So, for example, if X is a compact oriented dmanifold of virtual dimension k, and Y is a manifold, and f : X →Y is a 1morphism, then we can define a virtual class [X] in the homology group H_{k}(Y;ℤ), which is unchanged under deformations of X,f. This will be important in applications of dmanifolds and dorbifolds.
Many important areas of mathematics involve "counting" moduli spaces of geometric objects to define enumerative invariants or homology theories  for instance, Donaldson and SeibergWitten invariants for 4manifolds, DonaldsonThomas invariants of CalabiYau 3folds, GromovWitten invariants in algebraic or symplectic geometry, instanton Floer cohomology, Lagrangian Floer cohomology, contact homology, Symplectic Field Theory, Fukaya categories, ... In all of these (at least over ℂ for the algebraic geometry cases, and away from reducible connections, etc.) the moduli spaces concerned will be oriented dmanifolds or dorbifolds, and the counting can be done using virtual cycles or chains. So, dmanifolds and dorbifolds provide a unified way of looking at these counting problems. There are truncation functors from geometric structures currently used to define virtual classes to dmanifolds and dorbifolds. For instance, any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold is a dmanifold. In algebraic geometry, a ℂscheme with a perfect obstruction theory can be made into a dmanifold. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of Jholomorphic curves induce dorbifold structures.
In 2014 I found an alternative way to define derived manifolds and derived orbifolds, based on FukayaOhOhtaOno's Kuranishi spaces. I wrote up my new definition of Kuranishi spaces in arXiv:1409.6908, surveyed in arXiv:1510.07444, and am now writing a series of books on "Kuranishi spaces and Symplectic Geometry", which you can read about on another page on my website. These define an ordinary category μKur of 'μKuranishi spaces' which is equivalent to the homotopy category Ho(dMan) of the strict 2category of dmanifolds dMan, and a weak 2category mKur of 'mKuranishi spaces' which is equivalent to dMan, and a weak 2category Kur of 'Kuranishi spaces', which is equivalent to the strict 2category dOrb of dorbifolds. They also define boundary and corner versions μKur^{b}, μKur^{c}, mKur^{b}, mKur^{c}, Kur^{b}, Kur^{c}. These will be equivalent to Ho(dMan^{b}),dMan^{b},dOrb^{b},Ho(dMan^{c}),dMan^{c},dOrb^{c} for the versions of dMan^{b},...,dOrb^{c} in the eventual OUP book, but for technical reasons are not quite the same as those in the 2012 preliminary version.D. Borisov, 'Derived manifolds and Kuranishi models', arXiv:1212.1153.
D. Borisov and J. Noel, 'Simplicial approach to derived differential manifolds', arXiv:1112.0033.K. Fukaya, Y.G. Oh, H. Ohta and K. Ono, 'Lagrangian intersection Floer theory  anomaly and obstruction', Parts I and II. AMS/IP Studies in Advanced Mathematics, 46.1 and 46.2, A.M.S./International Press, 2009.
K. Fukaya and K. Ono, 'Arnold Conjecture and GromovWitten invariant', Topology 38 (1999), 9331048.
D. Joyce, 'Algebraic Geometry over C^{∞}rings', to appear in Memoirs of the A.M.S.D. Joyce, 'Dmanifolds, dorbifolds and derived differential
geometry: a detailed summary', arXiv:1208.4948, 2012.
D. Joyce, 'Dmanifolds and dorbifolds: a theory of derived differential geometry', preliminary version of book (2012) pdf file. 768 pages.
D. Joyce, 'A new definition of Kuranishi space', arXiv:1409.6908, 2014.
D. Joyce, 'Kuranishi spaces as a 2category', arXiv:1510.07444, 2015.J. Lurie, 'Derived Algebraic Geometry I: Stable ∞categories', math.CT/0608228, 2006.
J. Lurie, 'Derived Algebraic Geometry V: Structured spaces', arXiv:0905.0459, 2009.
D.I. Spivak, 'Derived smooth manifolds', Duke Mathematical Journal 153 (2010), 55128. arXiv:0810.5174.