Mathematically...


...speaking, these days I'm trying to understand singularities, which are quite "bad" mathematical objects. So I need to understand representations. Obviously, for this purpose I need to understand singularities first...

It seems that Noncommutative Geometry (NCG) provides a natural and useful framework to address the above problem(s). "Bad" mathematical objects (e.g. singularities) appear way much more often than the "good" ones, but the usual geometric methods are inadequate to understand them (by default, this what "bad" means). So one tries to replace such objects with suitable operator algebras, reflecting their "badness" to the noncommutativity of the product involved. Once this is done, NCG allows to do Geometry in this new context. Obviously, finding appropriate operators on these "bad" objects is the main issue here. Quite often, groupoids take care of this succesfully; modelling a bad space on a suitable groupoid transfers the "badness" to the symmetries built into the groupoid structure. Convolution then translates these symmetries to the noncommutativity of the associated algebra of functions. Groupoids also play a crucial role in Mathematical Physics, where one can find lots of singularities.