Pizza Seminar Talk: Tuesday, April 9, 2013, 4:30-5:30 PM, MC 108.
Speaker: Mitsuru Wilson
Noncommutative Geometry (NCG), as created by Alain Connes is a potent
mixture of many ideas in mathematics and physics.
One of the grand themes in mathematics, geometry, dates back for thousands of years. Starting from Euclidean spaces, many elementary shapes such as circles, disks, spheres have been studied until today; all spaces we learn in school are still studied today! Riemann then generalized Euclidean spaces with his foundation of manifolds, which was then used by Einstein for his theory which successfully encodes our knowledge of space-time and gravity.
With all they knew they did not understand quantum mechanics. The geometry is very fuzzy, uncertain, probabilistic and entangled there. Although it was introduced theoretically, this analogy works to explain noncommutative geometry (NCG) very accurately. The celebrated 1943 paper by Gelfand and Naimark proves the anti-equivalence (functors are contravariant) between (the
category of) compact Hausdor spaces X and (the category of) commutative C* algebras A. In the construction, A is nothing but C(X), the algebra of continuous functions . This correspondence gives rise to the viewpoint that any C* algebra is the space of functions C(X) of some noncomutative space X . They are fuzzy spaces in the sense that we cannot directly see them! My goal in this talk is to present the basic idea of Noncommutative Geometry in as elementary terms as possible and discuss a few canonical and fascinating examples in NCG.
Absolutely no background is necessary!