User feb7 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:11:41Z http://mathoverflow.net/feeds/user/3879 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14518/applications-of-noncommutative-geometry/14526#14526 Answer by Feb7 for Applications of Noncommutative Geometry Feb7 2010-02-07T18:20:53Z 2010-02-07T18:20:53Z <p>I do not think NCG arose as a way to solve problems in algebraic geometry using new methods. The motivation seems to be to broaden your horizons further.</p> <p>The answer of Bischof to the question you have cited, gives many contact points with classical topics in algebraic geometry such as deformation theory, invariant theory, moduli spaces, etc..</p> <p>Also see <a href="http://www.neverendingbooks.org/index.php/pollock-your-own-noncommutative-space.html" rel="nofollow">this article</a> of Lieven le Bryun, in which he speaks of points that can "talk to each other" via common tangent information. In other words, the "Chinese Remainder Theorem" fails for noncommutative rings and so points can be exceedingly close to each other. This is a more interesting way of looking at more general spaces. </p> <p>I suggest that you look more formally at the "noncommutative torus", which is a very important example in NCG. This space is a quintessential example of the above property of points being very close to each other.</p> <p>Then again, with the more abstract topics in algebraic geometry, n-categories, stacks and all that stuff, these developments could be carried over to noncommutative geometry, and since NCG is at the heart of many developments in physics, it might give wonderful applications to string theory etc., and in a deeper understanding of our physical world.</p>