There is a school of thought that suggests first contact with aliens should be on a purely mathematical basis so that we can prove to them we're not entirely ignorant angry apes. There would be no common language or social cues and gestures which could form an initial form of dialogue, not even pointing is as intuitive as we think it is.
With this in mind I would like a leading mathematician or physicist to be the earth representative. This still presents problems, as again, our numbers will not translate into an alien language. But if this representative if prepared, they would be able to demonstrate theorems symbolically. For example by drawing a right angled triangle and communicating that we understand the Pythagoras Theorem, or by demonstrating Pi. Mathematical communication of this nature has/is being attempted, for example the 1974 Arecibo broadcast and I find all of this stuff fascinating.
As you wrote, communication via mathematics only serves to show we're not entirely ignorant damned dirty apes and there is no dialogue. Wouldn't any demonstration of our understanding of ratios serve the same purpose? I don't think we need a skilled mathematician for that.