Advice for Berkeley Ph. D. students in math

Below are some suggestions based on my personal experiences at Berkeley first as a Ph. D. student and later as a professor advising students. Many of them are simply common sense, and go without saying, but I'll say them anyway.

• Preliminary exam. Take it every time you are eligible for it, even if you're not sure whether you're ready for it. You have three semesters to pass it, not three tries, and numerical scores are not recorded in your file available to professors.
• Language exams. Take them as soon as you can, starting with the language you know best (among French, German, Russian). You must pass at least one before attempting the qual. Most people find that it's not necessary to take a language course in order to pass the exam. To study, memorize the translations of some basic mathematical words (set, integer, homomorphism, continuous, open set, etc.), and learn some of the grammar, at least enough so that you know where the verbs are likely to be found! Practice by translating pages from foreign books from the math library (e.g. Bourbaki, for French).
• Useful skills. Early on, learn how to use LaTeX and MathSciNet. Learning BibTeX, or better yet amsrefs, also can save you a lot of time.
• Choosing an area and finding an advisor.
  • Go to expository seminars (Mentor lectures, MSRI-Evans lectures, the Department Colloquium, Many Cheerful Facts) to learn about areas of current research. Also go to seminars in areas that interest you. There will always be talks that are incomprehensible, but you'll eventually understand more of them if you keep going. Often it is the fault of the speaker, and you can sometimes improve the lecture for everybody by asking the speaker to explain something. So ask questions. If you are interested in number theory, add yourself to the number theory seminar mailing list.
  • Talk to professors; don't expect them to seek you out. Ask them what they are working on, how many students they already have, and what background material you would need to study in order to work with them. You can also go to MathSciNet and check out their recent papers.
  • Go to the library and browse recent articles in the best journals (such as Annals of Math., Invent. Math., Journal of the AMS, Duke Math. J., Math. Annalen, Internat. Math. Research Notices, etc.) If you see an article that looks interesting though you don't understand it very well, bring it to a professor in the subject area and ask for suggestions for background reading.
  • Consider taking a reading course with a potential advisor, to get a sense if he or she is a good fit for you. This can also be a good way to prepare for the qualifying exam, assuming you make the subject of the reading course one of the three topics of your qualifying exam.
  • Go to tea!
  Once you have a potential advisor, your future course selections and qualifying exam plans should be discussed with that person instead of your initially assigned graduate advisor.
• Qualifying exam.
  • Timing. The qual should be taken within 25 months of entering the program. This means that if you entered the Ph. D. program in a fall semester, you need to take it no later than the September at the beginning of your third year. At least six weeks before the exam, a draft of the syllabus should be prepared, the committee should be selected, and the syllabus should be distributed to all faculty in the two sections containing the topics on the syllabus. (See the graduate assistant in 910 Evans for details.) The committee should have four professors, including one who is an Academic Senate member (roughly, tenure-track or above) from a department other than mathematics, and you should ask in advance to make sure that at least one is willing to accept you as a student if you pass. About four weeks before the exam, you should formally apply to the exam, in 910. Go to 970 to reserve a room for the exam, and tell the graduate assistant in 910 the exact time and place.
  • Syllabus. You need to talk to your potential advisor, and decide upon three topics. The first should be the general area in which you want to do research, such as "algebraic number theory". The second is often a more specialized topic within (or related to) the first, such as "the arithmetic of elliptic curves", although some people simply choose a second general area, (e.g. "algebraic number theory" and "algebraic geometry" could be the first two topics). The third topic should be in a completely different section of mathematics. In the examples above, one could add "functional analysis" or "complex analysis" or "differentiable manifolds", for instance. Within each of the three topics, you should list the specific concepts and theorems to be covered. Look at the binder in 910 containing syllabi of former students. Most people simply cut and paste topics from these.
  • Committee. The committee consists of four professors (five for foundations). At least 50% must be from the mathematics department, and at least one must be from another department. To find an outside member, you may have to knock on a lot of doors in other departments. In order to pass the qual, at least one member of the committee must be willing to accept you as a student. Therefore you should make sure, long before you take the qual, that one of the committee members is willing to accept you, provided that you pass. It might still be possible to choose a different advisor later on.
  • Mock qual. Organize a group of advanced graduate students in your area who can unofficially administer a practice qual to you, preferably at least two weeks before your actual exam.
• Letters of recommendation. Plan in advance whom you will ask to write letters. Ask long before the letters are due, say a month in advance: this makes it harder for them to say "I'm too busy," and gives you time to find others if they cannot write a letter for you. Ask the letter writers to notify you when they have sent their letters. If the deadline is approaching and you haven't received notification, send a reminder to the letter writer. Also check with the recipient to make sure that the letter was actually received.
• Teaching recommendation. If you are a teaching assistant this semester, and are not sure if you will teach again before your final year, then ask the professor of the course to attend one of your discussion sections and write a letter of recommendation about your teaching. If your professor is unavailable or unwilling for some reason, maybe ask one of the Vice-Chairs for Undergraduate Affairs. It's better if the professor comes unannounced instead of on a pre-arranged date.
• Advertising yourself.
  • Make a professional webpage and keep it up-to-date, at least when it comes close to the time to apply for jobs. It may be easiest to copy someone else's webpage and then edit it.
  • If you write any papers, post them on your website and on the arXiv server.
  • If your paper is closely related to papers written by certain professors elsewhere in the world, mail a preprint to them. Or at least email them to let them know that it can be downloaded from your webpage. They might write letters of recommendation for you someday.
  • When you have results, apply to give a talk about them at a conference in your field. For example, the AMS organizes frequent meetings, notably the national meeting each January: each has "special sessions" on various topics, and anyone can apply to present a talk at one of these.
• Applying for jobs. As for academic positions, different institutions offer different types of positions, but there are some general patterns.
  • Major research universities (places with Ph. D. programs) tend to offer temporary postdoctoral positions, usually for up to 3 years. Almost all of these positions involve teaching, although the amount can vary; a typical load might be two courses each semester. At the end of such a job one is usually expected to apply for a tenure-track position elsewhere. Only in exceptional cases do research universities make tenure-track offers to new Ph. D.'s.
  • Primarily undergraduate colleges are more likely to offer tenure-track positions to new Ph. D.'s.
  • Some research institutes, such as MSRI and IAS, offer semester-long or year-long postdoctoral positions with no teaching duties. There are similar institutions in other countries. If you receive a one-year offer from one of these institutes and a three-year offer from a university, say, often you can negotiate with the university (before accepting) to defer their offer for a year.
  • Organizations such as the National Science Foundation, the American Institute of Mathematics, and the Clay Mathematics Institute may offer postdoctoral fellowships that can be held at various institutions.
  • Mathematics jobs outside academia. For instance, Microsoft Research hires a few mathematicians, and some national governments hire many mathematicians (though for the latter you may have to be a citizen of the country). These places also may offer summer jobs, even for students.
  • Other. Well, who wouldn't want to hire a smart mathematician?
  The same job title can have different meanings at different institutions: For instance, an "assistant professor" position may be tenure-track, or may be a temporary postdoctoral position. At a few places, such as Princeton, "assistant professor" positions are technically tenure-track, but the tenure rate is so low that for all practical purposes they are temporary positions. You can apply for as many as you have the time or the inclination for. (When finishing my Ph. D., I applied to about 50 schools, and I knew some people who applied for over 200; admittedly, the job market was very tight that year.) Most applications are due towards the end of the fall semester in the academic year before the job begins; the earliest deadlines are typically around October (though you should check this yourself). Therefore plan to have most of your application materials ready by the end of September. An application for an academic position at a university or college typically consists of the following:
  • Curriculum vitae. If you're not sure what should be included in a C.V., look at examples on mathematicians' websites to get a rough idea.
  • Teaching statement. Discuss your teaching philosophy: What do you think makes a teacher effective? How do you make the subject matter interesting for students? Are there techniques you tried but later abandoned because they did not work well? There are not many right or wrong answers here; the point is to show that you have given the matter thought.
  • Research statement. This describes your results, but more importantly discusses your ideas for future research. If possible, the first paragraph should be understandable by any mathematician; later it can get more technical. Try to give some background of your subject, and mention some previous work done by others if that helps to put your results in context. Explain why your research will be important, at least for the development of the mathematical field, if not for outside applications.
  • Abstract of your dissertation (possibly). It's not expected that you will have finished your dissertation by October. But it should be possible for you to state the main theorems, even if the proofs are not written down yet. (Of course you should not claim that you have proved things that you don't yet know how to prove!)
  • Letters of recommendation. You will probably need at least three, and at least one of those should address your teaching.
  There are many more good suggestions on the MGSA site and in the article on pages 1021-1026 of the October 2006 Notices.
• Publishing an article. Here are some things you might consider when choosing a journal to submit an article to.
  • You might look for a journal in which articles on a similar topic to yours have been published, and try to judge yourself whether your article is of similar quality.
  • Look at the list of editors of the journal (usually this can be found online, or at the front of each issue of the journal), and email your article to the one who is most likely to be interested in your article, following the journal's particular instructions.
  • The most prestigious journals may be Annals of Math., Invent. Math., and the Journal of the AMS. Others of very high quality include Duke Math. J., Math. Annalen, Internat. Math. Research Notices, J. reine und angew. Math., and Compositio Math.
  • Some journals are much more expensive than others: see the following list. Cheaper journals may be subscribed to by more libraries. Some people feel that boycotting overpriced journals is the right thing to do, to send a message to publishers.
  • Some journals have a backlog, which means that articles, even after they are accepted, may take a year or longer to appear, because they have accepted more articles than they can fit in current issues of the journal. The Notices of the AMS publishes a list of current backlogs in one of their issues each year.
  • Go to the library and see if you like the look of a journal. Some really do look better than others!