Summer Research at UMass
The department has its own Summer Research Program supported by gifts to the department and grants to individual professors: Research Experience for Undergraduates at UMass. Students accepted into the program typically work with individual professors on a project. Summer 2020 applications are due on March 10.
National REU programs
Students who are US citizens or permanent residents are encouraged to apply to one of the national REU programs in the mathematical sciences.
It's also possible work with faculty during the semester on a research project. This could be informal, or part of an independent study, or as the basis for an Honors Thesis.