Now

that my non-geometry

post is linked via the comments in this

string-coffee-table post which in turn is available through a

trackback from the Kontsevich-Soibelman

paper it is perhaps useful to add a few links.

The little

I've learned from reading about Connes-style non-commutative geometry is

this : if you have a situation where a discrete group is acting with a

bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole

complex-plane, rather than just the upper half plane) you can associate

to this a $C^*$-algebra and study invariants of it and interprete them

as topological information about this bad orbit space. An intruiging

example is the one I mentioned and where the additional noncommutative

points (coming from the orbits on the real axis) seem to contain a lot

of modular information as clarified by work of Manin&Marcolli and

Zagier. Probably the best introduction into Connes-style

non-commutative geometry from this perspective are the Lecture on

Arithmetic Noncommutative Geometry by Matilde Marcolli. To

algebraists : this trick is very similar to looking at the

skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if

you want to study the _orbifold_ for a finite group action on affine

space. But as algebraist we have to stick to affine varieties and

polynomials so we can only deal with the case of a finite group,

analysts can be sloppier in their functions, so they can also do

something when the group is infinite.

By the way, the

skew-group algebra idea is also why non-commutative algebraic

geometry enters string-theory via the link with orbifolds. The

easiest (and best understood) example is that of Kleinian singularities.

The best introduction to this idea is via the Representations

of quivers, preprojective algebras and deformations of quotient

singularities notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka

non-commutative projective geometry originated from the

work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in

the east) to understand Sklyanin algebras associated to elliptic curves

and automorphisms via ‘geometric' objects such as point- (and

fat-point-) modules, line-modules and the like. An excellent survey

paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby

Stafford and

Michel Van den Bergh. The best introduction is the (also

neverending…) book-project Non-

commutative algebraic geometry by Paul Smith who

maintains a

noncommutative geometry and algebra resource page page (which is

also available from the header).

Non-geometry

started with the seminal paper ‘Algebra extensions and

nonsingularity', J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim

Cuntz and Daniel Quillen but which is not available online. An

online introduction is Noncommutative smooth

spaces by Kontsevich and Rosenberg. Surely, different people have

different motivations to study non-geometry. I assume Cuntz got

interested because inductive limits of separable algebras are quasi-free

(aka formally smooth aka qurves). Kontsevich and Soibelman want to study

morphisms and deformations of $A_{\infty}$-categories as they explain in

their recent

paper. My own motivation to be interested in non-geometry is the

hope that in the next decades one will discover new exciting connections

between finite groups, algebraic curves and arithmetic groups (monstrous

moonshine being the first, and still not entirely understood,

instance of this). Part of the problem is that these three topics seem

to be quite different, yet by taking group-algebras of finite or

arithmetic groups and coordinate rings of affine smooth curves they all

turn out to be quasi-free algebras, so perhaps non-geometry is the

unifying theory behind these seemingly unrelated topics.