I will admit that I have been wondering what counts as “research”.
One daughter is involved in something that clearly counts as research. However, bringing it to completion would require millions of dollars (perhaps more like hundreds of millions) and perhaps ten years. Even how to do something similar as a PhD thesis seems hard to me.
There may however be things that a high school student could do to apply math to the real world. I have been trying to think of some (I was a math major).
I have done such simple things as use the Pythagorean Theorem to lay out a 90 degree angle in my vegetable garden (actually to create the outline of the garden in the first place). On the job I have used math for a variety of tasks, but it is hard to think of something that would be useful for a high school student. I suppose that there are things like statistics to determine whether a political poll is valid or not. Probability and statistics is however relatively advanced math, and is not something that I would have studied in high school.
One could compute how much CO2 is taken out of the atmosphere in a year by a small stand of trees, and try to plant enough trees to offset your yearly release of CO2. Figuring out how much CO2 is needed to create your food might not be easy, but what your home and your family’s cars release directly might be easier to figure out. Similarly you could compute how much CO2 is needed to mine the materials to build the batteries for a Tesla (plus the copper wire for the engine), find out how electricity is produced where you live, find out how much CO2 is released in the production of electricity versus burning of fossil fuels to run a car, and figure out how long it would take given the actual sources of electricity current used in your location to offset the CO2 released in producing the electric car and make it come out as less polluting compared to for example just driving whatever cars you actually have in your family.
Could you use calculus to compute the trajectory of a stone thrown by a catapult, build the catapult, and see how it compares? You would need some way to figure out what the direction and velocity of the stone would be at the point when it leaves the catapult. You might expect that friction (which is very difficult to take into account) would result in a shorter trajectory compared to what is computed, then try to figure out whether this is what occurred.
I do not know if any of these count as “research”. They might be applications of mathematics.
Another thing that comes to mind is trying to find an internship at a local company. This might be tough for a high school student interested in mathematics.
And learning a programming language and programming a robot to run around your house also comes to mind as a possibility.
One thing I noticed…
Does this mean that you are not in the USA? I am not sure that this matters in terms of anything that I wrote above. However, it might impact your chances of admissions to top schools in the USA.