This document was prompted by many discussions with graduate students about feeling lost as to how to transform from a new graduate student to ABD, and from ABD to PhD.
We hope that, in writing this information down in one place, we can help future graduate students feel more secure in how they chart their course toward their dissertation. Of course each student's experience will be individual; however, knowing the general layout of the land, so to speak, may help a student plan better what to do and what to expect.
N.B.: It is fine if you deviate from this outline. Everyone's experience is different, especially between Math and GGAM students. For example, it is normal not to take your quals until your third year (although you should pass them as soon as you can!) Also, you might have to wait to find a professor to work with; they might be on sabbatical, or too busy to take on any new students. The point is that you need to make progress. You should develop your own outline, tailored to your own research goals (and go over it with your advisor, why not?) that takes into account what your goals are.
N.B.: This document is written from the viewpoint of students who intend to stay in academia. Let this note be a plea for someone knowledgeable in obtaining jobs outside of academia to revise this document.
Introduction. Your first year will be hard but in many ways the most crucial. You should have two main goals: find an advisor or research area (details below) and pass the prelims. There are many required courses for first years and it is essential you keep up with them! It may seem overwhelming to take required courses while trying to break into research and coping with TA duties. The core courses are necessary for your breadth--an essential quality for any mathematician.
The last (and best) piece of advice we can give: talk to senior students. Many are more than willing to talk about anything grad school-related. If you're frustrated with life and/or work, talk to them--they've been there too. They also might give you some advice that will help you out down the road.
Courses and reading courses. During the first year, you will be taking many courses, some required. In the Winter and Spring terms, try to set up a reading course with other graduate students in an area that you are interested in knowing more about.
If you're not sure what to take a reading course in, attend some seminars and if a talk strikes your interest, try to find out more about that area. If you come in with a few different ideas of what you'd like to do, this is especially important--choose one and explore it. The fact that you are interested in many things should not prevent you from choosing one of them to examine in more depth.
Here is a nice post by a current graduate student at Berkeley that sheds insight into why your mathematical preferences may shift as you spend more time in graduate school (even if only subtly).
RFGs and seminars. Attend at least one research seminar and RFG. Give at least one talk in an RFG or student-run seminar. Seminars are a fantastic way of getting exposure to different fields. N.B.: Seminars, for the most part, are not classes, and shouldn't be treated exactly like classes. Here are some ways in which seminars are like classes: you should attend them, you should take notes if it helps you process the information, you should try to pay enough attention to get the big picture. Here are some ways in which seminars are not like classes: you should not expect to understand every word, and you shouldn't try (too hard) to fix that either. However, you should be able to answer the following questions at the end of any seminar: "What is the major theorem or question that the speaker is approaching?" "Why is he or she interested in this question or theorem?" (More on attending seminars.)
Initial foray into finding an advisor. During the spring, begin thinking about who you would like to work with. Talk to this person about what background you should begin acquiring. (You should come in with some ideas about this.) Work out a timeline for gaining this background.
Summer research. Do a research project with your potential advisor over the summer. At least once every two weeks, write down what you have learned (from papers, books, conversations) or discovered (through your hard and persistent work). Keep this in a place that you will be able to refer to later. (See "A word of encouragement: integrating epsilon-sized progress over time".)
Pass the prelims.
Planning for quals. Discuss with your research advisor when to take the quals, what should be on your qual exam, and who should be on your quals committee. (You should come in with suggestions for all these.) Draw up a timeline for quals prep and follow it as best you can.
Continue your research project. (See "A word of encouragement".)
Regular maintenance of your mathematical breadth. Continue attending seminars and reading papers. Read papers in your area that seem relevant to the problem that you're working on. Don't worry about the details of the paper unless you genuinely need to know them. Do work out examples and be able to answer the question, "what is the main point of this paper?" Give at least two talks in RFGs and student-run seminars.
Attend at least one conference in your area.
Pass your quals. (Qualifying exam advice page.)
Regular maintenance on your research project. By this point, you should be able to articulate a very rough plan of attack for your research project (I will use X technique to do Y to show Z). It doesn't matter if this plan turns out to be wrong. A plan provides you with direction, a necessary condition for finishing any project. Even if in the course of pursuing this plan, you find it's wrong, that's progress! Just make sure to come up with a new plan.
Regular maintenance of your mathematical breadth. Continue to attend seminars and read papers in your area. Attend at least one conference in your area. Give at least two talks in RFGs or student-run seminars.
Teaching duties, with an eye toward future academic appointments. Be an AI for at least one course, and make sure that your mentor or some other faculty you trust watches you teach. If you're unhappy with the way your teaching that day goes, then find another day. Why: if you go on in academia, you will need a teaching letter. The letter will be far stronger if they can say that they've seen you teach.
N.B.: If you plan to graduate in Year n, you need to be very close to finishing your project this year.
Regular maintenance of your research project and mathematical breadth. Continue doing the activities of Year 3. If you plan to graduate in Year 5, you should be able to write an abstract of your thesis by the end of Year 4.
Ideally, you should have a preprint out by spring of Year n-1. If you do, or you are close, then begin talking to the people that you've met at conferences about speaking at their schools' research seminars.
Networking with other mathematicians. Attend at least one conference in your area. (At least!) if you have any results, talk to other people about them, and try to give an MAA or AMS sectional talk on your work.
What if you are interested in liberal arts jobs? If you think it is a possibility that you will want to take a liberal arts job or teaching job after you graduate, then attend the Joint Math Meetings and begin meeting people from the types of schools you are considering a job at.
If you haven't yet, then be an AI at least one course. Think carefully about who you would like to write you a teaching letter (this person should be a faculty developer, lecturer, post-doc, or professor). Have that person attend the course. If you are unhappy with the way that day went, then find another day for them to come. (But talk to the person first--you may find that you were overly harsh in assessing your own performance.)
Apply for fellowships. It will make your life more pleasant if you are able to travel next year without having to arrange for your section to be covered.
Getting your research project together for the job search. If you don't have a preprint yet, then make one happen. But if you can't, it's not the end of the world: what is important is that you can describe coherently why your work is interesting and how it relates to questions that may interest other people (such as the people hiring you). It is also important that the relevance of your work be communicated to your letter writers. No matter what, get as much research done as you can, because you won't have much time in the fall.
Networking with an eye toward getting a job. Attend a conference in your field and tell as many people as possible about your result. Mention that you're graduating.
Preparing for the year ahead. Have an idea of where you will apply. Write your teaching and research statements. Trust us, you will appreciate having done this. Make a few drafts of cover letters if you have time.
Getting together job search statements: Summer-November. Revise your teaching and research statements as needed. Apply for the NSF or any other appropriate fellowships.
Obtaining letters of reference: September-November. Contact your letter writers at least 6 weeks before any deadline. (An insightful page on letter writing, by a math professor at Stanford, Ravi Vakil.) You will need at least 1 teaching letter and 3 research letters. Ideally, you will have an external letter writer. Some liberal arts and teaching schools will require more than 1 teaching letter.
N.B.: The NSF deadline is mid-October. This means that you will need to contact letter writers in the beginning of September!
If you don't adhere to this strictly, you'll probably be okay, and it's still worth asking for letters. Keep in mind that whoever is writing your letters will be writing other letters and that letters take time to write.
Organizing your job search: August-October. Make a list of all the places you plan to apply with deadlines, requirements, etc. Especially if you apply for liberal arts and teaching jobs, you will need to keep track of a dizzying array of individual job application requirements.
Show your advisor your list and talk to her/him about whether this list is reasonable. Get a second or third opinion if you feel uncomfortable with what your advisor says (or doesn't say).
Advertise your result: August-January. Send your preprint to people who may be interested in your result--ask them what they think of it. Try to arrange to speak at conferences and other schools' seminars on your result. Do a practice run at one of the research seminars here. Talking to other people about your result is especially important if you do not have preprints and publications on recent work.
Turn in your applications: October-January. Watch out for deadlines--there are a few deadlines before November. Try to submit applications early. If you can't, or worse, are late, it may be worth turning in the application anyway.
Joint Math Meetings: January. Especially if you are interested in liberal arts positions, attend this conference. Many schools conduct interviews there. If you are interested in post-doctoral research job, it may be worth going to give a talk so that you can network with the people running the session in which you speak.
Wait for and decide upon job offers: January-May, possibly June. Talk to your advisor and others about the choices that you receive. Do some soul-searching. If you are on the tenure-track market and are so lucky as to receive multiple job offers, make sure to talk to your advisor and as many other people as reasonable about negotiating strategies and goals.
Find out dissertation deadlines: whenever you have time to figure this out, before March. Make sure you know what the filing deadlines are that year. The university is very strict about these, so make sure you have the dates straight.
Write a first draft of your dissertation: Early as possible-March. Turn a first draft--with complete sentences, figures, and references-- to your committee by March 31. Expect to have to revise--a lot.
Revise the first draft: March-end of May. Read every line. Critically. You knew this already, most likely. You will find mathematical and grammatical errors, and because the dissertation is such a long document, you should expect to catch new errors for at least the first three to five times you go through your draft. You will want to rearrange material. This process will take a while. There's a passage that all students go through in revising the dissertation: you begin by rewriting three (or more) pages at a time, then you revise three paragraphs at a time, then you revise three sentences at a time, and finally, you revise three words or letters at a time.
Format your dissertation: last two weeks of May or earlier. Grad studies has a number of requirements. Whether or not you think they are reasonable, your diploma depends on your adherence to them. (UC Davis Grad Studies Filing requirements) (TBA: Link to Galois Group thesis LaTeX template.)
Buy dissertation paper: April-May. Sometimes stores run out. Buy early.
Get your committee to sign your dissertation: May. If anyone in your committee is on sabbatical, you may need to FedEx documents to the Ukraine, or wherever they are. This takes extra time, so plan ahead.
File your dissertation. Do it on time. Really. The university makes no exceptions, and if you don't file on time, you won't get your diploma on time. Read all the requirements, twice. Adhere to each one strictly. Try not to file on the last day possible.
At some point in graduate school, every student is told to attend seminars. However, the purpose of seminars may be a bit mysterious--after all, it is nearly impossible to understand every word (or even every third word) for the entire hour. This section is placed here to give a potential answer to (1) why, despite not seeming to understand them, you should still attend seminars, (2) what you might do during a seminar to help keep yourself paying attention.
The material here is taken (and slightly edited) from Professor Ravi Vakil's page, with permission.
Think actively about the creative process. A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class you may have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?
Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.
When you learn the theory, you should try to calculate some toy cases, and think of some explicit basic examples.
Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.
You probably will not solve the problem you have set out to work on over the summer or over the year, even if you have planned to do so. That's the nature of research. Just keep plugging at it--research has a surprising way of taking dirac-delta-function-like jumps when you are persistent, even if at any moment, it may not feel like you're making huge progress. (... apologies for the mathematical inconsistency of the metaphor between the title of this section and this last sentence. Nonetheless--) It is therefore very important to be persistent in your work ethic, because (1) it is difficult to measure progress at any particular moment, and you must slog through that, and (2) you will never see any gains if you do not work. No matter what you do, it may help to work out lots of examples and calculations so that you can build intuition, and to keep the big picture at the forefront ("What problem am I working on?" "What approach am I taking to it?" "Where does this approach work and where does it seem to fail?" "Why would people in the area be interested in this problem?").
It is natural to feel stupid sometimes. Here is an article that speaks to "importance of stupidity" in research. I found this article very encouraging as well as insightful.