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, $GL2(\mathbb{Z})$ 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 $\mathbb{C}[x1,\hdots,xn] * G$ 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.

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Artin, arxiv, Connes, Cuntz, geometry, groups, Klein, Kontsevich, Manin, Marcolli, modular, moonshine, non-commutative, noncommutative, Quillen, quivers, representations

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Posted in geometry
Written on Wed, 21 June 2006 at 3:41 pm
Tags: Artin, arxiv, Connes, Cuntz, geometry, groups, Klein, Kontsevich, Manin, Marcolli, modular, moonshine, non-commutative, noncommutative, Quillen, quivers, representations
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