The federal government funds many programs where undergraduates can study mathematics and other sciences. Among these are the REU programs (Research Experiences for Undergraduates), usually offered for 8 to 10 weeks in the summer. Here is a list of current REUs in mathematics. Oregon State has had an REU in mathematics for over twenty-five years.
A successful applicant has usually completed several courses that are similar to our 300-level courses in analysis or algebra, but that is not always the case. A very helpful article by Bill Velez (U of Arizona) collects the REUs according to how much experience is required.
Applications are typically due by the middle of February. The application requires letters of reference written by faculty members who know your work. Be sure to confirm their willingness to write well in advance of the deadline.