The research into the existence or non-existence of dark matter definitely needs a lot of mathematical analysis, like many other branches of science. It probably doesn't need new types of mathematics, and, if it did, any new mathematical axioms required would have no direct bearing on the existence of dark matter, only on the mathematical structures used. Think of the maths as being akin to the screws in a telescope metaphorically speaking - the telescope couldn't exist without them, but the principles of the construction of those screws has no direct relevance to the discovery of new astronomical objects. To give you a more substantial parallel, Einstein used tensor maths to formulate the general theory of relativity. Tensor maths was obscure at the time, but it didn't need any new mathematical foundations because that had all been done by a mathematician called Riemann about 70 or 80 years earlier. Riemann was working on the purely mathematical analysis of non-linear multi-dimensional geometries and the axiomatic foundations of his calculus would have rested in this field - he had no idea that it could be used to describe how gravitation works. Einstein used it to great effect, but his physics made no difference whatsoever to the foundations of the mathematical principles behind tensor calculus.
