See http://math.stuy.edu/summer-programs/ for an overview of the math programs listed below:

(please note that although NYC Math Circle is not listed in the above overview, it has been attended by Stuyvesant students and the teachers are highly experienced and well-respected math teachers and math team coaches).

Macaulay Honors College The Summer Scholars Academy in Mathematics and Science at CCNY https://summerscholarsacademy.theschoolsystem.net/ssa.html

New York City Math Circle http://www.nymathcircle.org/

Hampshire College Summer Studies in Mathematics http://www.hcssim.org/

PROMYS      http://www.promys.org/
Ross Mathematics Program    http://www.math.ohio-state.edu/ross/
Awesome Math Summer Program       http://awesomemath.org/
Canada/USA Math Camp       http://www.mathcamp.org/
The Texas Mathworks Honors Summer Math Camp
http://www.txstate.edu/mathworks/camps/hsmc.html

The following program is very prestigious and is actually more of a science program:
Research Science Institute (RSI)   http://www.cee.org/programs/rsi

Math With Mrs. Avigdor

Hampshire College Summer Studies in Mathematics http://www.hcssim.org/