Abstract

The answer to this question is simply that new measurements need new numerical systems and old numerical systems are not adequate as seen by the history of the development of the numerical systems, from Natural numbers to Integer numbers to Rational numbers to Real numbers to Complex numbers to Vectorial numbers. Thus new measurements of microscopic systems forced Heisenberg to reinvent the new numerical systems of matrices in which each new number was an infinite square table of complex numbers or equivalently linear operators on Hilbert space. By the work of Von Neumann and others, the theory of operator algebras became a rich theory, so mathematicians like A. Connes began to use it as a powerful tool in mathematics. The program of noncommutative geometry, like algebraic geometry and topology is to study a space through an algebraic structure which one assigns to the space very naturally, but this time the algebra is noncommutative and carries a topological and ordering structure that is an operator algebra. In this lecture we try to investigate some examples of spaces for which the classical tools of analysis like topologies, measures, met- rics, calculi will give poor results but new tools of noncommutative geometry will give good results.



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