SURF

The Summer Undergraduate Research Fellowship is a ten-week summer research program at Caltech, although some students opt to perform their research with a faculty member of another university. If you're looking to do a SURF in math, it doesn't usually suffice to mass-email the math faculty -- you should go speak to them in person. Try to explain your interests to them and maybe they or another faculty member they know can help you get started doing research. Sometimes it's hard for an undergraduate to know exactly what they want to do, but it often suffices to tell them that you find their research interesting and that you'd like to perform your own research in their field. Despite rumors you may have heard, freshmen can find research in mathematics, no matter their background, as long as they are determined. In general it is often difficult to get research in math because there's no lab to pay for it, so having a good proposal is key. Some example proposals can be found here.

REU

Another way that Caltech undergrads have gotten involved in mathematical research in the past is through the Research Experience for Undergraduates program funded by the National Science Foundation. There currently no REU at Caltech, so the program would take place at another college.