# Diffeology and Noncommutative Geometry

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Patrick Iglesias-Zemmour

created by daniele on 21 May 2018

7 jun 2018
-- 14:30

Aula Tricerri, DiMaI, Firenze

**Abstract.**

Noncommutative geometry has been invented to describe geometrical/smooth structures on “bad” quotients. Diffeology has been invented a priori to describe infinite dimensional groups, especially groups of symplectomorphisms. It happened that both met on a famous exemple: the noncommutative torus or irrational torus, that is, the quotient of a 2-torus by a line with irrational slope. Both descriptions have a common result: on the one hand the C*-algebra of two noncommutative tori are Morita-equivalent if the slopes are conjugate modulo GL(2,Z). On the other hand, two irrational tori are diffeomorphic also if the slopes are conjugate modulo GL(2,Z). That coincidence could not be just by chance. I shall show how we can understand this correlation, in a special case, by associating a C*-algebra to every diffeological orbifold in such a natural way that two diffeomorphic orbifolds give two Morita-equivalent C*-algebras. I will discuss some directions to generalise the construction.