Hot and timely research problem with a slow student

If you have taken on the responsibility of shepherding this student to her doctorate then you have an ethical constraint that overwhelms most other considerations. The only exception to that, that I can see, is if the research is actually life saving in some way.

I a lot of supervisor-supervisee relationships the advisor can do the work on her/his own without the student. But that doesn't ever imply that the student should be left behind. The supervisor has more experience in field and in research in nearly all cases, so the student requiring a (possibly long) learning curve is pretty typical.

The "hotness" of the research topic doesn't change your ethical responsibilities. If you don't want to honor that you shouldn't take on advisor roles.

However as aeismail suggests, the student needs a suitable problem, not necessarily this problem. If she is at the beginning of the process it might be reasonable to morph her participation into a different problem.

But, your assessment of her overall ability may also be wrong. She may, in fact, have a breakthrough moment in which things fall into place. And if the problem is that she lacks key background elements for some reason, you could make it go faster by finding ways for her to quickly come up to speed on those.

Finally, it isn't uncommon in many fields, including mathematics, for an advisor to give some hints to the student, pointing out the direction to be explored, if not more.

If she’s truly in the beginning phase of her PhD, it would be entirely appropriate not to wait, unless you have specifically defined the tasks she needs to do as an explicit part of her thesis research. In that case, you must proceed with extreme caution, because you are fundamentally changing things after everyone has agreed on a plan.

Otherwise, if she has already made some contributions, incorporating them into ongoing research and including her in the publication process would be a good way to help her develop her skills and motivate her to make further contributions because she can see directly what can be done with it. You can also mention that other results she gets in the near future could also be used.

However, as mentioned by others, if you are a subordinate co-supervisor, you are not able to follow this advice unilaterally and should consult with the principal supervisor to make sure you are in agreement on this strategy.