I agree with José's comment above: I do not think early specialisation is a good idea. Did I understand correctly that you want to give a one week to a 15-year old to decide on which area of mathematics to specialize?
I want to add something different, however. I fail to see how "some undergraduate topics currently taught compulsorily are a bit of a burden". Mathematics is not a set of disconnected areas. They are all highly related. Most research problems, while staying in one area, may be related to another, motivated by another, applicable in another, or steal ideas or techniques from another. One general course in, say, real analysis, complex analysis, abstract algebra, differential geometry, discrete mathematics, or topology is not a burden, but I dare say an actual necessity for anybody wanting to do research on any topic in pure math. To use your own example, someone doing research in Lie theory will benefit from, rather than be burdened by, a solid understanding of basic differential geometry. Or to use my own case, I am a Poisson geometer, but I have used ideas or results from all the above topics in my research.