Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv, A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes’ flavor is the most successful. Most of Connes’ examples are variants of the same basic idea: when a group acts nicely on a space, you can define a new space by collapsing each orbit of the group action to a single point (this construction is known as the quotient space of the action). Unfortunately, most group actions are not nice.
Connes and Marcolli describe an alternative construction. By a theorem of Gelfand, you can study spaces by instead studying its ring of continuous functions (see this Wikipedia article for precise details). Gelfand’s result puts the commutative in commutative geometry. For group actions that have badly behaved quotients, Connes introduced a noncommutative ring that functions as the analogue of the quotient space.
[...] For anyone who finds the Connes-Marcolli paper we recently linked to heavy going, Lieven Le Bruyn recommends some lighter reading: an interview with Alain Connes himself. [...]
[...] A few days ago, Ars Mathematica wrote : Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes’ flavor is the most successful. [...]
[...] A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over. [...]
[...] Alain Connes, who we’ve talked about before, has added something very exciting to his website: the text of his famous book Noncommutative Geometry as a pdf. There is nothing quite like Connes’ book. It’s part popularization, part research monograph, and part manifesto. I highly recommend taking a look. [...]