US PhD Admission Instructions

It so happened that I entered the PhD program in mathematics at Stanford University. I would like to talk about what is required for admission, and why to try to enter is not very difficult.

What is required for admission?

  • TOEFL ($ 260) (English exam)
  • GRE Subject Mathematics ($ 150) (math exam)
  • GRE general ($ 205) (school math and English)
  • At least three letters of recommendation
  • Statement of Purpose (essay)
  • Curriculum vitae (scientific autobiography)
  • (sometimes) Personal History Statement (other essay)
  • (preferably) publications and reports at conferences
  • Money (average $ 150 per university)

And now – in order!


TOEFL is an English language exam that consists of four parts: Reading, Listening, Speaking and Writing. A detailed description of the exam can be found on the official website. The maximum score is 120.

Instead of TOEFL, you can take the IELTS exam, which is also suitable for American universities. But I didn’t write IELTS and I don’t know anything about him, so I will continue to write about TOEFL.

As far as I know, for admission you need to score more than the minimum threshold, which each university has its own direction. After passing this threshold, the difference in points should not matter. For example, for the commission there is no difference between 100 and 120 points, but a result of less than 90 can cause problems. For most areas, 100 points is a sufficient result.

On the Internet you can find a lot of materials for preparing for TOEFL. However, it should be noted that the number of complete tests on the Internet is limited (especially with listening). Therefore, it is better to carefully study the errors in the test done, than immediately start the next one.

I used the following materials:

Cracking the TOEFL book contains a lot of good exercises and a couple of complete tests with listening. However, the text of the book itself is too detailed, and, in my opinion, it is not necessary to read it. It is better to simply solve problems along the text, returning to the explanations only when problems arise. The same goes for The Official Guide to the TOEFL.

Be sure to look at examples of good essays to understand what ETS experts are expected to see from you. Examples can be found in this file or on the official edX course (which I do not advise to take the whole course).

I was preparing for the delivery for about a month with a deep dive and got 108 out of 120. Many advise to devote more time to preparation, and this depends on the level of language proficiency. However, even for people with good English, it makes sense to prepare for an unusual test format. I did not study with a tutor, did not go to paid courses and used only materials that are available on the Internet. But I do not claim that this is the right approach.

GRE mathematics

GRE mathematics is a math exam. It has a lot of tasks and little time to solve them. 50% of problems in mathematical analysis, 25% in linear algebra, and the remaining 25% are different sections of mathematics. To successfully pass this exam, you must be able to quickly solve problems. Details can be found on the official website. The range of points is from 200 to 990.

On this site you can find exams from previous years with solutions. The Princeton Review has also released Cracking GRE Subject Mathematics, which contains many tasks, but again, the text is too detailed. It is better to solve problems and return to theory if necessary.

The result of this exam is one of the main factors for the selection committee. I scored a low score (760 out of 990) because I don’t know how to solve problems quickly (unlike those who at one time did olympiads), and this was the weakest point in my application for admission.

GRE general

The test consists of two parts: school mathematics and surprisingly complex English (test and two essays). On the official website you can find a full description and examples of the problem. The ranking is rather strange: from 130 to 170 for each part, and essays are graded on a scale of 1 to 6.

For a person with a technical education, school mathematics should be trivial, and the commission expects a result close to the maximum (170). English, by contrast, is incomprehensible (at least for me). Honestly, I do not know how to prepare and successfully pass this exam if you do not know literary English for many years. Complex tasks on the vocabulary made me literally guess the answer option without any reason. The Cracking the GRE book of the same edition of the Princeton Review did not help much.

On the other hand, as I understand it, this does not really matter. Probably, this exam is part of the bureaucratic system, and the result in English does not affect anything (in a reasonable sense). However, in mathematics, of course, you should score the maximum score.

All my friends who entered the United States scored an absurdly low score for the language part of this test. However, in other areas everything may be different (probably a high score is required for English literature).

Letters of recommendation

The main idea: “Give material to those who write a recommendation!”
If you are a good student, they may write to you about your studies.
If you speak at seminars – about academic activity.
If you do research, it’s about scientific results.

If there is nothing, then there is nothing to write about.

Letters of recommendation are best received naturally: to study well, to do non-trivial term papers, to speak at seminars and conferences, to discuss scientific results or research questions with professors. It’s not worth it to “knock out” recommendations to yourself, specially intersecting with famous scientists, if you do not have a common scientific topic. This is unprofessional.

One letter of recommendation should be from a faculty member where you study. Other letters of recommendation can be from anyone, but it’s right to choose those people who can tell something specific and at the same time – good.

Statement of purpose

Statement of Purpose is an essay where you need to explain why you and the chosen university are ideally suited to each other. There are many conflicting opinions about what needs to be written in the Statement of Purpose, and no one knows for sure what will work and what won’t.

I think that it is not worth writing about interests other than the direction of interest. If you’re an outstanding athlete, play in an orchestra, or speak six languages, it might be better to indicate this in your resume. An exception can only be if it fits perfectly into the text or a hobby is indirectly related to the chosen direction (programming for mathematics is quite good).

In my essay, I wrote about the journey that has been made, why I decided to do mathematics, what goals (or dreams) I want to achieve, and why the chosen university is an ideal place to realize them. I think it is appropriate to indicate specific people, if these are not randomly chosen names, but people whose results you know and are really interested in working with them.

It is important to show (if true) that you are ready to become part of the academic world or to indicate other goals that you pursue. In the end, universities expect their students to be great scientists and develop an academic environment: write books, teach and hold conferences.

Curriculum vitae

CV is the easiest part because it contains only facts and nothing more. The CV should include your education, olympiads, teaching experience, speeches at conferences, published works, knowledge of languages and, possibly, other interests (somewhere in the end).

By and large, CV is a short scientific autobiography.

Personal history statement

Some universities (for example, Berkeley) ask you to write another essay about the difficulties that you have overcome in your career. Problems with discrimination, prejudice or anything like that. I didn’t really understand what they want to hear, so I wrote rather casually that I changed institutes twice to be in Moscow, where there are more chances for a good career. I am not sure that this essay is generally important. Perhaps this is just a bureaucratic demand.

Publication / Conferences

Having research experience before applying for a PhD may be useful, but by no means necessary. Such experience shows that you are able to deal with scientific issues, and not just study well. In addition, scientific results will be the content of letters of recommendation and CV. However, in general, students are not expected to have any results at this stage and, theoretically, this should not seriously affect admission.


On average, a university application fee costs $ 100. But in addition to the collection, you need to send the results of TOEFL and GRE, which costs $ 20 and 27, respectively (however, the first four submissions are provided by TOEFL for free). Thus, on average, applying for one university costs $ 150.

You also need to pay for exams, which in total costs $ 260 (TOEFL) + $ 150 (GRE subject math) + $ 205 (GRE general) = $ 615.

What is the result?

Of course, for admission you need to make some efforts, but on the other hand, the admission process is linear and understandable. You can prepare and pass the exams well. Instead of a term or diploma, you can ask the research supervisor for a research question and speak with him at seminars and conferences. You can study hard and get a high GPA, which in many places is paid a lot of attention (especially if your university has established itself as a good education). And everything will turn out!



  • On December 20, 2019