Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An introduction to C-infinity schemes and C-infinity algebraic geometry   This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry
over C-infinity rings".
  If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions
c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n
--> R there is an n-fold operation \Phi_f : C^\infty(X)^n --> C^\infty(X)
acting by \Phi_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations
\Phi_f satisfy many natural identities. Thus, C^\infty(X) actually has a far
richer structure than the obvious R-algebra structure.
  We explain a version of algebraic geometry in which rings or algebras are
replaced by C-infinity rings. As schemes are the basic objects in algebraic
geometry, the new basic objects are "C-infinity schemes", a category of
geometric objects generalizing manifolds, and whose morphisms generalize smooth
maps. We also discuss "C-infinity stacks", including Deligne-Mumford C-infinity
stacks, a 2-category of geometric objects generalizing orbifolds. We study
quasicoherent and coherent sheaves on C-infinity schemes and C-infinity stacks,
and orbifold strata of Deligne-Mumford C-infinity stacks. This enables us to
use the tools of algebraic geometry in differential geometry, and to describe
singular spaces such as moduli spaces occurring in differential geometric
problems.
  Many of these ideas are not new: C-infinity rings and C-infinity schemes have
long been part of synthetic differential geometry. But we develop them in new
directions. In a new book, surveyed in arXiv:1206.4207 and at greater length in
arXiv:1208.4948, the author uses C-infinity algebraic geometry to develop a
theory of "derived differential geometry", which studies "d-manifolds" and
"d-orbifolds", derived versions of smooth manifolds and orbifolds. D-orbifolds
will have applications in symplectic geometry, as the geometric structure on
moduli spaces of J-holomorphic curves.

28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th