With admission to Research Experiences for Undergraduates being competitive, it is possible that undergraduate mathematics students may seek alternative summer plans. Fortunately, there are non-REU mathematical research and learning opportunities that can be as valuable as an REU. Examples include:
Independent studies or reading courses with a UVa faculty mentor. This can be done for credit (which may require paying summer tuition) or informally (which will not require summer tuition).
Summer research with a UVa faculty mentor. This has the possibility of being funded through an undergraduate research grant or a faculty mentor's grant, among other means. Ask your peers and faculty mentors for more information about funding.
Research residency at another research institution.
Summer semester abroad programs. For example, the Budapest Semesters in Mathematics summer program, which has a research component. Semester abroad programs typically require paying tuition, but credits do transfer to UVa and financial aid/scholarships may be available. See this website for more details on how to pursue a semester abroad program as a UVa undergraduate student.