Professor Daugherty's advice for
Applying to PhD Programs in Mathematics

Preparing

If you're starting to think about whether a PhD program might be right for you, or you're looking to apply in a couple of years from now, here are some things that can help you in the meantime:

• Courses: Take advanced courses, and do well in them. If this means taking fewer classes some semester so you can do really well in a special topics in algebra course or a graduate level analysis class, do it. If you're finishing your undergraduate degree now, and don't have any advanced courses on your transcript, you may consider supplementing your education first:
  • Get a masters degree: Masters degree programs come in a few different flavors, but broadly speaking, some of them are designed to prepare future PhD candidates and some are designed more to meet regional workforce needs (some are both). You're looking for the former. You can tell which is which usually by reading the program description, and looking at the course availability within the program (these should basically mirror the courses someone would take in the first 2 years of a PhD). Note that, as opposed to most PhD programs, masters programs might cost money. When applying, keep an eye on what tuition funding and stipends are available, and how long the programs (reasonably) take.
  • Do a "bridge" program: These are programs that are specifically designed to bridge the gap between a less rigorous undergrad degree in math, or a bachelor's degree in some other field, and what is needed to enter into a PhD programs well-prepared. They are often (but not always) geared toward underrepresented groups, and offer funding and stipends. Some of these are actually masters programs (like U Penn's Bridge to PhD 2-year program), some of these are supplemental semester or summer programs (like the EDGE summer program for women, or Smith College's Center for Women in Mathematics Postbaccalaureate Program). The MAA also keeps a list of semester and summer programs in math.
• Research: Seek out research opportunities.
  
  • For US citizens, there are several Research Experiences for Undergraduates (REUs) all over the country every summer. Application deadlines are spread out over the spring term. They pay you, and often offer housing. Some REU programs have other funding for non-US citizens as well.
  • Many math departments have internal funding for their own students to do research locally over the summer. Occasionally, professors will have their own funding through grants as well. Look for announcements soliciting applications, or talk to your professors about your interest in doing research so that they might be able to point you in the right direction.
  To approach a professor about working with them, first try to get an idea of the kinds of mathematics they work on by reading their websites or looking up some of their papers. As an undergraduate or masters student, though, don't worry about bringing them a specific problem to work on—the adviser usually provides a problem or subject for students at your level.
• Extra-curriculars: Look for experiences that will go beyond the standard undergraduate curriculum. For example:
  • Attend a summer school like those listed above, or like the Undergraduate Summer School at the Park City Mathematics Institute or Mathematics Advanced Study Semesters at Penn State.
  • Do a semester abroad in the Budapest Semesters in Mathematics.
  • Be a counselor for a high school math camp.
  • Find tutoring or outreach opportunities in math.
  • Learn a programming language (even if you're not planning to study computational mathematics).
  • Get involved with local math organizations, like math or STEM oriented clubs at your school, or local math or STEM oriented outreach programs (at middle and high schools, museums, etc.).
  • Study for and take the Putnam Exam. Your department or local math club may get together as a group to train; if not, consider starting your own group!
  The American Mathematical Society keeps sites for Programs for Graduate Students and Recent Ph.D.s and AMS for students (for high schoolers and undergraduates). The Society for Industrial and Applied Mathematics also keeps a list of Fellowship and Research Opportunities for undergrads, grads, and postdocs.
• Talk to your professors: To find out if the academic track is right for you, talk to other humans who have lived it. They can speak to how long it all takes, what is involved, what the job is like, what the life is like, what different kinds of jobs are out there for someone with a PhD in math. Also, eventually, you'll need some of them to write recommendation letters for you. So while you're taking someone's class, make a (good) impression: go to office hours, ask and answer questions, try to do any "bonus" problems that get assigned, and work on your mathematical writing in their courses (this stands out more than you think). See "Communicating Mathematics through Homework and Exams" and "Some Guidelines for Good Mathematical Writing". This training will also help you to write like a professional in several places in your application.
• Do your homework: Find out all you can about different graduate programs—What research is happening? How many PhDs do they graduate per year and where do those graduates get jobs? Are people (students and faculty) happy and productive there? Go visit, go to conferences and talk to people, read webpages, etc. There are rankings of math PhD programs out there, but more important to your success is finding an adviser who you will work well with and a community that will support you and help you start your career. Way more so than choosing an undergraduate institution, picking a PhD program is a complex matching program, with optimality being a function of place, resources, personalities, connections, etc.

GREs

Though many schools have recently transitioned away from requiring suject-specific standardized exams, some schools do still require that you take Graduate Record Examinations (GREs), both the general and subject tests. These are basically the graduate analog to the SATs. To do well (particularly on the subject test), you'll actually need to study, and perhaps take them more than once, so plan ahead. Different schools weight them differently in their assessment processes—some use them as hard cut-offs, some just use them as part of the bigger picture. When they're an optional part of your package, if you do well, they can help offset any negative irregularieties that you might have elsewhere in your record. The general GREs are being offered all the time, and can be scheduled online. The subject tests only happen a couple of times a year (September, October, and April at the time this is written—see Registration, Test Centers, and Dates for current information), and require planning significantly ahead of time. And while it's important not to mess up the general GRE too badly, the subject test is often the one that is more important to your application.

Paying for it: If taking these exams presents a financial hardship, ETS has a Fee Reduction Program. Your school or department may also have some funds set aside for helping with these fees.

Preparing for it: Information on the structure of the exam can be found both on the ETS site linked above, and on the Wikipedia pages on GREs and GRE Math. There are lots of sample problems online (like ETS's GRE Math Test Practice Book), and first- and second-hand GRE prep books for sale. Consider organizing a study group—get together every week or two, pick a subject area, and work problems together. Sign up for the next subject test now to give yourself a hard deadline (and also so you don't forget to do it later).

How good is good enough? Keep in mind that the majority of people taking the math subject test are other prospective mathematicians. While a few more points might make a difference (especially if other parts of your application are weak or unknown), a 50th percentile score on the math subject test is average amongst math PhD hopefuls, a very different scale than average amongst all PhD hopefuls (like the general GRE). As you start to make a list of programs to which you might apply, read their applications process page for any information on how they use GREs. If that information isn't there, you can email a couple of schools (whoever is listed as the math department contact for the graduate admission process) to ask what their expectations are with respect to GRE scores. (If you do this, as always, be formal, polite, clear, and brief.)

English exams: Foreign students will also often be asked to complete a standardized English exam, such as the TOEFL. Occasionally, an institution will require some permanent US residents or citizens to complete the TOEFL as well, so read the requirements for applications far ahead of time, and plan ahead if needed.

Letters of recommendation

In general, you'll need about 3 letters of recommendation. These should come from professors who best know your recent work. If you've done any independent studies or research work in the last couple of years, your mentor there is a great choice. If you've recently done well in an advanced course, and you're pretty sure your professor knew your name, that's another good choice. Your high school math teacher, your calculus professor, or your close relative who has an advanced degree are all poor choices. Professors who only know you from Fall semester might be ok, but aren't necessarily the best choice since they won't have time to really get to know your work before letters are due. And though there might be special cases, letters from active researchers with permanent positions tend to carry more weight than temporary adjunct faculty.

When requesting letters of recommendation, keep in mind that your professors are probably writing a lot of letters—for PhD programs, fellowships, postdocs, tenure reviews, etc. This means two things: First, don't be afraid to ask, since this is something that they do a lot—they might say no, or that they won't be the best choice for you, but it won't hurt to ask. Second, if they agree to write for you, be really respectful of your professors' time and provide them with all the resources and information that will help them to write that letter. They're probably swamped with lots of other tasks, in particular lots of other letters to write (and frankly, it's hard to write 20 personalized letters in a single application season without some help from you). Another word on asking: if a faculty member declines to write a letter or shows reluctance, that is a bad sign. All letters will be submitted directly to the institution—you will not be able to vet them ahead of time. So don't prod a reluctant letter writer; just find someone else.

Good etiquette:

• 4+ weeks before your first deadline, email and/or talk to your potential letter writers to let them know you would like to request a letter of recommendation for PhD programs (or REUs or whatever). Potentially set up a meeting to discuss the process with them. Be polite, respectful, organized, and efficient.
• 3+ weeks out, provide a list of schools you're applying to with
  • deadlines;
  • how to submit letters, including email or web addresses or pre-addressed envelopes;
  • links to relevant application instructions;
  • digital and/or physical copies of forms they need to fill out, or links to online forms;
  • relevant information to help them write the letter (how they know you, how you did in their course, a current CV, unofficial transcripts, a draft of your personal statement, etc.).
  For example, here is the list of information that I require from anyone for whom I write letters.
• 4–7 days before the deadline, check the status of letter submissions with your first school if possible. If it's not in, don't be alarmed—professors run on very busy schedules, and it might just be slated as a task for close to the due date. Send a brief, gentle, non-accusatory reminder that the first application deadline is coming up to make sure the task hasn't gotten lost in the shuffle. You can do this under the guise of asking if there's anything else they need.
• When you get accepted somewhere, and especially when you decide where you're going, let your letter writers know! It makes us feel good to know we helped you. If you don't get into a school you want to attend by the end of the season, you should still let your letter writers know—they might be able to help you plan the next step. Either way, consider giving them a thank you card to express your appreciation of the time they take on your behalf.

Personal statement or "Statement of Purpose"

This 1–2 page document is your opportunity to let the admissions committee hear why you want to go to get a PhD, in your own voice. Your GREs, your transcripts, and your activities show your preparation and history of commitment; your letters of recommendations reflect the impression you've made before on professionals in the field; but your statement allows you to talk about your vision of the future and how your experiences shape that vision. In that light, the two major points you want to address in this statement are

1. why you want to get a PhD, and
2. what you plan to do with your PhD.

You can (and should) also use your statement to highlight your experiences, such as research projects, summer schools, advanced courses, etc. But the point isn't just to regurgitate your CV or resume. Use the discussions of these experiences as an opportunity to show your maturity and investment (how you talk about your experiences is important here).

As one of my own mentors put it, getting a PhD is a slog. Everyone, even amongst the brightest minds in mathematics, thinks about quitting at some point (for me it was during a summer school at the IAS; and then, again, every time I had to apply for academic jobs). The people who succeed are the ones who have internal safety nets—things that will keep them invested, keep them going through the hard times. When an admissions committee is sifting through several hundreds of applications, they're looking for signs that an applicant will be successful (i.e. finish, relatively happy, with a PhD). So as you're writing and editing, try to put yourself in their shoes: What can you say to demonstrate that you have some idea of what you're getting into? What can you say to show them that you're going to be successful in their program? This is also your chance to explain anything about your path that is not evident elsewhere (like if there have been difficult obstacles that you have overcome, this may be your moment to positively portray personal assets that don't otherwise appear, or even which look like poor performance elsewhere on paper).

Tone: Be explicit, sincere, positive, and concise. Don't be self-depricating. Be mindful of your audience (senior mathematicians evaluating your potential success in mathematics). Be professional, but not overly dry—let the reader know who you are, without inadvertently offending a stranger.

Process: Start early. Have your friends and mentors read your statement for you, and edit, edit, edit. Make an appointment (or several) with the your school's writing center.

The Math Alliance has a great page on this topic, with several links to other helpful resources.

Other stuff

Course descriptions: Keep a list of every math class you've taken since starting college, annotated with text book, final grade, and a summary of the topics covered. Many applications require this information in some form.

Make your application easy to navigate: Read the application directions carefully at each school, and follow them. Use standard file types, and consider generating pdf files to ensure nothing happens in transit (if you are not using TeX, you can "print to pdf"). Include your name and document type (e.g. personal statement) on every page, perhaps in a header or footer on secondary pages.

Funding expectations

It is reasonable to expect that a PhD program admission will include an offer of tuition waiver and stipend for your first 5 years. The stipend is typically linked to you working as a teaching or research assistant. However, institutions may have variability in their budgets to support graduate students from year to year. One benefit of applying to several institutions is increasing the odds that you have options when it comes to accepting, and can take funding into account. (If applying broadly presents a financial hardship, find out if the institutions have fee waiver programs; for those that don't, your school or department may have some funds for helping to reimburse application costs for students.)

Your letter of admission will outline the finances associated with your offer. Even if you feel like "Wow! Any money is more money than I had hoped for!", once you have such a letter, talk to other people (professors and friends) to get a sense of how fair or practical the offer actually is. If an acceptance doesn't come with a stipend, but you are specifically interested in the institution because there was someone with whom who you wanted to work, consider contacting that person before rejecting the offer—they may have grant money to support graduate students, or ideas of how to find funding for you.

The AMS also keeps a list of Awards, Fellowships, and Other Opportunities, which you can filter by audience (i.e. graduate students). This includes scholarships to cover or supplement your tuition or stipends, as well as travel grants etc. The process for applying for external support also takes a lot of time, and should be started early.

More resources

Mathematical Association of America (MAA): the national professional society focusing on mathematics accessible at the undergraduate level. They keep a great page on programs for students, including semester and summer programs, conferences, research opportunities, etc.

American Mathematical Society (AMS): another national mathematical society for furthering "mathematical research, scholarship and education". They keep a page on AMS for students, which has information about math programs, research opportunities, careers in mathematics, etc. They also have an e-Mentoring Network, with blog posts about various issues that come up all along the mathematical career.

Math Alliance: a program centered around the goal of ensuring that "every underrepresented or under-served American student with the talent and the ambition has the opportunity to earn a doctoral degree in a mathematical science." They have a network of mentors who can help you through the process of applying and moving through a PhD. They also have events like their annual Field of Dreams Conference, with grad school and REU fairs, and panels on preparing for PhD programs. Their Grad School Prep page is excellent, and includes more instructions and advice for applying to PhD programs.

Graduate Research Opportunities for Women (GROW): "a conference for women-identified students interested in graduate school in the mathematical sciences." Every Fall at Northwestern University; travel expenses and meals are covered by them. Northwestern also has a general Resources for Undergraduate Mathematics Programs and Grants page with several great resources.

"How to Email Your Professor": This is an article specifically about emailing someone who is teaching a class that you're taking, but the general form will also serve you well for emailing potential letter writers or graduate admissions committee contacts. For contacting people who you don't know (like at other schools), do an internet search for them first to find the appropriate title to use in addressing them (their "honorific").

Sample PhD applications

Columbia
Dartmouth
Oregon State University
University of Minnesota
University of Wisconsin, Madison
CUNY Graduate Center

Thanks to the many other professors who gave great input and feedback in developing this page, including Sean Cleary, Brooke Feigon, Elizabeth Niese, Arun Ram, Ben Webster, and others. For corrections or updates, please email Professor Zajj Daugherty directly.