It is well known that a compactly supported function is smooth iff its Fourier transform is quickly decreasing. The classic theory of Hörmander generalizes this fact in terms of the so-called wavefront set. It can be defined for any distribution (i.e. any element of the topological dual of a the space of compactly supported smooth functions). It encodes not only local information but "micro-local" information, i.e. not only the singular support (where the distribution fails to be smooth), but also the directions where the (localized) Fourier transform does not decrease quickly, hence the cause for a singularity to appear in the first place. Using the wavefront set, it is straightforward to formulate a sufficient - and yet powerful - criterion when two distributions can be "pointwise" multiplied. I will explain the fundamentals of these constructions and point out how a generalization of Hörmander's wavefront set can be used to give an existence criterion for a certain class of other products, the so-called twisted product and twisted convolution product of a smaller class of distributions, the so-called tempered distributions. I will briefly motivate why such products are interesting in mathematical physics, e.g. when defining quantum field theory on a model of a noncommutative spacetime (the so-called Moyal plane). No background in physics is required.

Einladung von Prof. Anke Pohl