Topic: this is a mathematics education question (but applies to other sciences too).
Assumptions: my first assumption is that most mathematical concepts used in research are not intrinsically more complicated to grasp than high-school and undergraduate maths, the main difference is the amount of prerequisites (and hence time and experience) involved. My second assumption is that some undergraduate topics currently taught compulsarily are a bit of a burden for someone focussed on a particular topic.
Now of course cognitive development is a constraint, but upon reaching the age of high-school, I would think that a fairly large proportion of the scientifically-enclined students could really understand things usually taught much later and indeed become active at research level within a few years, provided some shortcuts are introduced.
Early specialization: I'm wondering if a balanced curriculum already exists (or is planned) to provide such early specialization. What I'm looking for is this: a one-week panorama of maths (or physics, or biology) would be organized at the beginning, and then the students would decide which subtopic to study. For example someone interested by group theory (or quantum optics, or genetics) would thus start with basics at the age 15 or 16, and gradually learn more stuff and skills, but for a few years with a strong emphasis on things directly relevant for the chosen subtopic.
So for example the student specializing in group theory would only learn differential calculus and manifolds in passing in the context of Lie groups, and would skip most undergraduate real and functional analysis until it becomes relevant for his/her research topic, if at all. Of course other general courses would still be taught (history, sciences, programming, foreign languages…), but at least 50% of the student's week would be devoted to the research topic, ensuring satisfying progress.
Question: do you know of any active or planned educative curriculum (at a high-school or university, or maybe a specific home-schooling program) as outlined above? As an example of successful early specialization see e.g. the winners of the Siemens Foundation Prizes, but I haven't been able to learn much about their specific curriculum if any.
Note: Skipping grades in school to enter university earlier is not the point, I'm really interested in a subtopic-oriented curriculum.
Best Answer
As far as getting high school students involved in research by learning rapidly a narrow range of mathematics but in some depth, this is actually done in the mathematics section of the Research Science Institute program at MIT for students who have completed their junior year. Last year there were four projects in representation theory; I recall that one of them did not now linear algebra until some two weeks before the program (but learned quickly and completed a very successful project).
Sadly, I do not know of many other opportunities- RSI is a small program, and only a portion of it is for mathematics. I believe the PROMYS program supervises some research projects, but it is primarily for learning mathematics. Incidentally, many of the winners of competitions such as Siemens begin their projects at RSI.
Also, alumni of the RSI program do not necessarily end up specializing in the same fields that they did their projects (if they do eventually choose to pursue a career in mathematics, which does not always happen). It does give an exposure to a certain field, though.