Some mathematics majors choose to participate in the summer REU programs (Research Experiences for Undergraduates) available across the country each year. For a list of such programs see these web sites:
The Mathematics Department also supports a small number of mathematics majors doing summer research here in Princeton for a total of 8 weeks during the summer. This program is open primarily to students who will be junior or senior mathematics majors during the following academic year; it is an opportunity not just to learn a body of mathematics but also to engage in some independent investigation.
A student wishing to participate must find a faculty member who agrees to supervise the program, and in consultation with the advisor must submit an outline of the proposed program via SAFE by the deadline (February 22, 2019 for Summer 2019). (Note that finding housing is up to the student.) Early in the spring semester, a reminder will be posted in upper division course web pages with more details about submitting an application.
Limited funding is also available for off-campus summer research. Application consists of a description of the program, the amount of funding requested as well as possible other sources of funding. The deadline for these requests is February 22, 2019 for summer 2019 programs; the applications should be submitted via SAFE.
From time to time the Mathematics Department has additional summer mathematics programs for undergraduates. These are announced on the Mathematics Department web page.