Advice for UMass undergraduate students pursuing research in pure mathematics

  • What follows is my personal opinion which is not endorsed by UMass. In recent years I've talked to a few talented math majors who finished the formal requirements of the pure math concentration but were not accepted into their desired graduate program in pure mathematics. I've decided to write down ideas that I always pitch to my mentees, and will hopefully also be useful to other students. It goes without saying that in order to succeed you need to be talented and hard-working, but there are also many resources available to you that are often overlooked. In addition to years of mentoring experience, this advice is based on reading and evaluating graduate school and fellowship applications.

    • Get a PhD

  • If you are passionate about pure mathematics and want to pursue a career in academia, the best pathway to do so is to attend one of the top graduate programs in the field. The current academic job market is tough and, for better or worse, most vacancies in pure mathematics are filled by graduates from a few elite universities. These schools offer not only excellent research and employment opportunities, but also an incredible study environment. You learn from never-ending discussions with your classmates, maybe even more so than from the classes and the faculty.
  • A successful graduate school application is a combination of an excellent GPA, a very good GRE score, strong upper-level coursework, research experience, undergraduate awards and honors, and a strong commitment to teaching and community outreach. Your application should be supported by three faculty letters clearly demonstrating your potential for graduate studies.
  • All of these elements of an application are important. Don't put all of your eggs in one basket by overly focusing on, for example, research experience at the expense of academic excellence. You only have a few years to conquer this optimization problem, so start now.
  • Explore your options. For some students, a good Masters program will work better than a PhD program. Some students find it advantageous to study mathematics for an extra year on their own before applying to a PhD program. Other interesting options include postbaccalaureate programs or working or volunteering in an education-related field (Teach for America, Fulbright, ...).

    • Coursework

  • The undergraduate math curriculum at UMass is not designed specifically for students pursuing careers in academia and/or research in pure mathematics. (Many of our students pursue other concentrations, such as teaching or computing.) The department does offer a few (mostly honors) undergraduate courses that are tailored specifically to students interested in pursuing research in pure mathematics, but these courses generally do not teach any late 20th century or 21st century mathematics. There's a certain competitive disadvantage in math preparation compared to certain elite universities, which do offer more advanced courses in pure mathematics to undergraduates. (Transferring to an elite undergraduate program is very difficult and happens very rarely. You have to prove that you are truly extraordinary (for example, enroll in Math 611 or 621 in your freshman year and be the best student in class - amazingly, this has happened).) However, this disadvantage can be overcome through a combination of taking graduate courses and self-study.
  • The UMass math department offers "core" graduate courses that are well-designed, often taught by top faculty, and always have quite a few strong undergraduate students. The material these classes cover is similar to the advanced undergraduate curriculum at elite universities, so these courses are great to have on your transcript. Strong students can finish the algebra, analysis or topology sequences (or even several sequences and sometimes even topics graduate courses on top of core sequences) by the time they apply for graduate programs. In my experience, several undergraduate students every year are sufficiently prepared to even bypass some of the undergraduate courses and jump directly into the corresponding graduate courses. If you are one of those students who find "upper-level" mathematical courses insufficiently challenging, you can try to waive them and take graduate courses instead (611-612 instead of 411-412, 623-624 instead of 523-524, 621 instead of 421). It will be a good idea to talk to the professor scheduled to teach the graduate course about course expectations and requirements. You can also always ask your current professors about your level of preparation for graduate courses. It doesn't make much sense to take a graduate course only to fail it or to trail other students. Waiving an undergraduate course requirement is not guaranteed and has to be discussed with the Chief Undergraduate Advisor.

    • Mentoring network and Research Experience

  • Find faculty mentors to help you navigate the maze of academia. Never hesitate to ask your faculty advisor and your professors for advice. Start developing mentoring relationships early and cultivate them over the years. Be an active participant in your classes, ask questions, and show your curiosity and passion for math. Recommendation letters are probably the most important part of a graduate school application and the best letters are written by people who know you well and can comment about your progress over years of study.
  • Don't be shy about approaching faculty members for reading courses and independent studies. The department has many research-active faculty willing to work individually with undergraduate students. Check the NSF DMS website to see which faculty are supported by NSF grants - they are research-active by definition and may even have summer funding available to work with you on a project.
  • The department has an excellent REU program funded by generous alumni contributions. Apply for this program (and for multiple outside REUs) starting in your sophomore year. This is the most straightforward and organized way to gain research experience and to develop mentoring relationships. Join the Honors College and write a senior thesis with your faculty mentor.
  • Join the Math Club and attend research seminars and colloquia - ask your mentor about the most inspiring and accessible talks you could attend. Regional research conferences often have activities accessible to strong undergrads.

    • Personal Statement

  • Graduate programs and fellowships typically don't expect you to pursue the same area of mathematics you studied as an undergrad. However, they will use your personal statement as a gauge of your preparation, enthusiasm, and ability to learn new things. Make sure everything you write in your statement is mathematically precise and ask your mentor to read it. Don't try to be original, be authentic. Talk about the most advanced mathematical theories you have learned. We all love Fibonacci numbers, but focusing on something elementary at the expense of more advanced topics may backfire. From the perspective of graduate admissions, they are looking for evidence that you are capable of learning new and difficult material over many years of graduate study.
  • To gain teaching experience, grade homework for the Math department and become a UGTA.

    • Exams and Awards

  • Start preparing for the GRE exams early. The department offers GRE preparation workshops that you will find useful.
  • Take the Putnam exam, win the JCK competition, apply for NSF GRFP and various scholarships (Goldwater, Churchill, ...)

    • Working with me

  • If you are interested in working with me, you can read an informal description of my research interests and check out the work done by my students. I generally prefer to work with students who have taken at least Graduate Algebra 611, but I can make an exception for students who come from ethnic groups, families, and/or regions that are underrepresented in graduate study in mathematics. I will be recruiting students in January 2022 and January 2023 - please send me your application (CV, unofficial transcript and statement of interest) in December if you wish to be considered. We would do an independent study in the spring, followed by an NSF-funded REU in the summer, followed by an Honors thesis in the fall.
  • If you have any further questions or concerns, I will be happy to answer them personally or by e-mail.