Ideally this refers to either original research or contribution to research through calculations/programming work. It is not uncommon to those who participated in a research program to culminate into a publication, but know that these are not necessarily large results.
Sometimes independent study of a recent paper/report is also called undergraduate research. Nonetheless, publishing is rare among undergraduate students who have not participated in such a research program.
Make sure you have a strong foundation in coursework (calculus sequence, linear algebra, proof class (discrete math or number systems)). Beyond this, the more mathematical maturity (an internal understanding that you gain but cannot be taught) will give you an edge. Of course, more coursework helps: abstract algebra (groups, rings, fields and special topics), analysis (real, complex, functional, measures), topology, combinatorics, number theory, numerical analysis, partial differential equations, and more.
And as we enter more deeply into the computional world, some programming is often desired (Matlab, Python, C++, etc).
Find other like-minded students and hold your own weekly or bi-monthly study group/seminar in which you take turns presenting things to each other. These could be short articles of interest found by flipping through MAA journals (or even read refereed/scholarly articles), surveying one specific topic, or working through some appropriate book. These are often called Math reading groups.
If you ask for a letter of recommendation, contact your potential letter writer at least 4 weeks prior to the deadline. Writing a good letter can be quite time consuming, and many faculty are understandably busy. When someone writes a letter of recommendation for you, they are being asked to comment not only on your past or current performance but -- most importantly -- your promise for future accomplishments.
Once they have agreed, give your letter writer everything they need to write and submit your letter (your future goals, CV, unofficial transcript, what semester/timeline you took did coursework/research with them, any other relevant infomation including deadlines). Ideally you can send this in one email. In general, it is best to get a request a letter from faculty members (and not graduate students, though they might have taught your course).
After the letter has been submitted, it is a nice gesture to send a thank you note to the person who wrote you a letter. After you receive a response from the program(s) that you applied to, it is also polite to let your letter writers know the outcome.
Besides having a high GPA and stong letters of references, it is also required to take the Graduate Record Exams (GRE). There are two types of GREs: general and subject. The general test is required and administerd online up to three times per month and has three components: verbal reasoning, quantitative reasoning, and analytical writing). The Math subject exam is non-trivial but is not required (but suggested) for many universities.
Juniors: It is best to take the mathematics Graduate Record Exams (GRE) subject test in your junior year. The subject GRE in Mathematics is a paper based exam. It is adminstered locally on dates in September, October, and April. See the GRE website for details and a practice exam booklet.
Seniors: If you plan to study mathematics in graduate school, and if you haven't yet taken the GRE, sign up immediately to take it in the fall. Start thinking about applications for graduate school early in the fall. Application deadlines typically vary between December and January.
- Most top-ranked graduate programs typically prefer a cummulative GPA of 3.5 or better. Exceptions to this rule occur, of course, but many students give up their quest to attend graduate school due to a low (3.0 or less) GPA.
- Most REU application deadlines are in February (of Junior year), but it's important to start early and not wait until the last minute. Winter break is a good time to work on applications, but most applications should be up by November and it never hurts to start thinking about them earlier. Attend successful summer REU between Junior and Senior year.
- In the summer of Junior year, start researching graduate school and take practice general/subject GRE.
- Take general/subject GRE in September of Senior year.
- Write you Statement of Purpose/ Letter of Intent (answering who/what/why questions) and request official transcripts from your undergraduate institution by October.
- Retake general/subject GRE by November, if needed.
- Complete and submit all applications by early December, verifying that your recommendation letters have been sent.
If you are a declared Math major or minor, you should automatically be on our mailing list. (You can always unsubscribe if you pick another major.) You can also sign up here. We will use this list to let you know about math club talks, opportunities for research opportunities and internships, and other important information for the motivated major.