Advice on how to start a Research (maths related)

Wow 4HLs!
German is likely to help if you are aiming for German-speaking universities - Tu München, ETZH?

For math, thinking a bit out of the box: what about NYU (Courant institute) and Williams?

yeah you got me, ETZH and Tu München are my dream choices in Europe
as for NYU, despite the prestige, the rental fee might be too high in New York, so I’m not sure if it’s worth it

You wouldn’t be renting. College students live in college residences (“dorms”). You’re right they’re very expensive in NYC and for NYU - but the comprehensive fee includes tuition&fees (cost of classes, using library&labs, office hours, support centers, health centers, athletic facilities…) + room and board (a room in a college residence, shared with another student the same age as you and generally same gender unless you request differently, sometimes sharing academic or extracurricular interests if you join “living learning communities” + access to 3 meals, with unlimited food, each day.)
Williams includes these costs, as well as the cost of books, transportation, and clothes, when they determine how much it’d cost your family then how much they need to give you for you to attend.
NYU doesn’t “meet need” so you never know how much they’ll decide to give you -often, it’s not enough to pay for the university costs, also known as a soft rejection.

what is the annual cost for NYU? Is it more expensive than MIT for instance?

The issue isn’t annual cost but available financial aid.
You can look up current (comprehensive) costs by typing “cost of attendance”+ name of any college. From that page you should see “net price calculator”. Or you can type the name of the college+ “NPC”.
The NPC will not be accurate - you have to run it as if you were a domestic applicant, which you’re not, so any federal aid wouldn’t apply - but it’ll help you understand the amounts+range involved and see net costs at a variety of universities.

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