Undergraduate research at UW Madison

CURLs (Collaborative Undergraduate Research Labs)
This link has a list of upcoming Undergraduate Research Labs. These labs offer opportunities for undergraduate to do research with Faculty members and postdoctoral fellows in the Mathematics Department.
Independent Research
Some faculty members do independent research projects with undergraduates. If interested, please contact the Undergraduate Advisor, Alexandru Ionescu in 309 Van Vleck.
Research Funding
You can also obtain financial support from the University for research. See this list of Research Grants and Programs

 Research Experience for Undergraduates (REUs)

NSF REU site
REUs are funded by the NSF. This site explains the national program and has the official list of REUs running each year.
Madison REUs
This is the list of future planned REUs to be offered in Madison

 Women in Mathematics

Two of the best programs for Women undergraduates in Mathematics are:

Carleton College
Summer program for freshmen and sophomores
George Washington University
Summer program for juniors and seniors
They are both well established, have run for many years and are very successful.

 Other Links

There are many other opportunities to attend a research program as an undergraduate. Here are some links:

American Mathematical Society Web Page
with links to REUs but also summer camps and more.
University of Nebraska Research page.
This site has a very complete list of opportunities, have a look!
Undergraduate Research
with links to many undergraduate research opportunities in Math and Physics. Among them there is an interesting one in the AT&T labs (for applied mathematics).
Carnegie Mellon
Summer program at the CMU Applied Math Institute.
Minnesota
This is an awesome program run by the faculty of the University of Minnesota at Duluth. It has been in place for over twenty years and some of our best undergrads have gone there.