Why is the Poincaré conjecture important?
The interesting thing with conjectures is that you can just assume they’re right and do all the work that would have followed it anyway. You don’t need to wait until it’s actually proved to build upon it. Of course your new work isn’t proven until the conjecture is, but that doesn’t really hold anybody up.
Knowing the answer to Poincare’s conjecture won’t build you a better toaster oven tommorow. There’s no way that not knowing the answer to a “yes/no” question (which is what Poincare is, really; “Is everything that with the same algebraic data as a sphere necessarily a sphere?” “Yes,” as it turns out.) can really impede technology. If some technology really depended on knowing the answer to Poincare you’d just go ahead and build it, assuming it was true, and if your gadget failed unexpectedly, well, then you’d have learned something. So the primary benefit of Perelman’s work is not the answer to the a big question, but the techniques he developed in the process of tackling it. The machinery of Ricci flows almost certainly will benefit applied mathematicians in a lot of unanticipated ways, and mathematicians and engineers who could not possibly have refined the machinery the way Perelman did can still use it now that he has. That’s the practical benefit of big, “pure” problems like Poinc
