Research experience
At some point in your undergrad math journey, you may want to get a taste of what math research is like. If you are considering doing a PhD in math, doing an undergrad research can be very beneficial. It is a good opportunity to test out whether you might enjoy grad school or a particular area of math. Below we attempt to address a few commonly asked questions.
Q: What kinds of research experience is meaningful, either for personal development or for grad school application?
No one expects undergrads to produce groundbreaking or even nontrivial results. Especially for pure math, a good research experience could be an independent reading project on an advanced topic culminating in an expository paper, or working out an interesting example of a complex theory in full details. Even though you are not proving anything new, you are honing important skills for actual research, such as reading difficult materials and developing your perspectives on a particular topic.
Q: How does one find reseach opportunities?
Unfortunately, research opportunities in math are not as advertised and organized as in some other departments. One standard route is to apply for summer programs at MIT (SPUR, SPUR+, UROP+...) and elsewhere (e.g. REU, a listing of which can found here). These programs are often highly organized, so it might save you some trouble finding projects. Most of them are mentored by grad students, which can be a plus since grad students are less intimidating and more likely to invest time into the projects. Applications ususally open at the beginning of the year, so keep an eye on the relevant websites.
Another route is to approach instructors of the courses you enjoyed and ask if they are willing to supervise a UROP, preferrably towards or at the end of the term. In general, postdocs are more willing to supervise a project during the semester, but it varies a lot.
A last resort is cold emailing, which is often a hit-and-miss. It might be a good idea to do some "research" about the faculty members you are contacting and, if possible, ask around to see if they are approachable and friendly.