Advice for 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 at Berkeley
and MIT advising students.
Many of them are simply common sense, and go without saying,
but I'll say them anyway.
 Preliminary exam (for Berkeley students).
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.
 Useful skills.
Early on, learn how to
use LaTeX
and MathSciNet.
Learning BibTeX or amsrefs
can save you a lot of time too.
 Choosing an area and finding an advisor.
 Go to expository seminars
(Mentor lectures, MSRIEvans lectures,
Department Colloquium, Many Cheerful Facts at Berkeley;
Grad Student Lunch Seminar, Joint Math Colloquium, PUMA GRASS, SPAMS
at MIT; etc.)
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 lists (nt and STAGE at MIT).
 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 the professor's website,
arXiv, or
MathSciNet
to check out their recent papers.
 Go to the library and browse recent articles in the best journals
(some are listed later on this page).
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 (for Berkeley students, though some of the advice is general).
 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, tenuretrack 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 at least a month before the deadline;
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.
See these suggestions
on how to request a letter of recommendation.
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.
 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
(or another professor)
to attend one of your recitation sections and write a letter
of recommendation about your teaching.
It's better if the professor comes unannounced instead of on a
prearranged date.
 Advertising yourself.
 Make a professional webpage and keep it uptodate.
It may be easiest to copy someone else's webpage and then edit it.
 If you write a paper, try to get feedback on it from someone,
perhaps your advisor. Once it is polished,
post it on your website
and on the arXiv server.
Wait a few days to see if you get any comments,
and then submit it to a journal.
 If your paper is closely related to papers written by certain professors
elsewhere in the world, 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,
and there are the Joint Mathematics Meetings each January:
each has "special sessions" on various topics,
and anyone can apply to present a talk at one of these.
 If you are invited to give a talk somewhere, generally you should accept!
 Publishing an article.
It is very helpful to have articles accepted for publication
before job applications are due.
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., Publ. Math. IHES, Annales de l'ENS,
J. Europ. Math. Soc., Geometry & Topology,
Math. Annalen, Compositio Math., J. reine und angew. Math.,
Amer. J. Math., Internat. Math. Research Notices,
and (though I may be biased) Algebra & Number Theory.
Other highly regarded journals include J. Algebraic Geom., Adv. Math.,
Math. Res. Letters, Trans. of the AMS, J. Inst. Math. Jussieu,
Israel J. Math., and some London Math. Soc. journals.
(Note: This is not intended as a comprehensive list or as a ranking.
The standings of various journals change over time, different people
have different opinions, and different journals are strong in
different subjects.

Some journals are much more expensive than others.
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.
On the other hand, what matters is when a paper is accepted,
not when it physically appears in the journal,
so don't worry much about the backlog.

Go to the library and see if you like the look of a journal.
Some really do look better than others!
Within a few days of submitting an article,
the journal should send you confirmation that it was received;
if not, email the journal.
If you receive no further news about your submission after six months,
it is OK to email the editor to politely ask what the status of your
submission is.
 Applying for jobs.
Timing:
 Most applications are due during the fall semester
in the academic year before the job begins.
 Some distinguished fellowships may have deadlines in September or October.
 Most academic positions have deadlines in November or December.
 You should send your application materials to your recommendation letter writers at least one month before the deadline.
Given all the above, it would be wise to have your application materials
ready by September at the latest.
Types of positions for new Ph.D.'s:

Research postdoctoral positions.
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 also 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 tenuretrack
position elsewhere.
As a new Ph.D., one would not normally apply to tenuretrack positions
at a research university that also offers postdoctoral fellowships,
unless explicitly asked to apply for such a position.

Research institutes.
Some institutes, such as
MSRI and
IAS,
offer semesterlong or yearlong postdoctoral positions
with no teaching duties.
There are similar institutions in other countries.
If you receive a oneyear offer from one of these institutes
and a threeyear offer from a university, say,
often you can negotiate with the university (before accepting)
to defer their offer for a year.

Distinguished fellowships.
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.

Tenuretrack positions at primarily undergraduate colleges.
Such colleges are more likely to offer tenuretrack positions to new Ph.D.'s.
These positions will definitely involve teaching;
some may also expect some research.

Special teaching positions.
A few research universities offer special postdoctoral positions
that involve only teaching (e.g., Harvard's "preceptor" positions);
these are intended for excellent teachers
who plan to go on to teaching positions at undergraduate colleges.
There are also organizations such as BEAM and
MIT Open Learning,
which may have positions.

Industry jobs.
Microsoft Research, Google, and other companies hire a few mathematicians,
especially those with programming experience or with interests in
computer science.
These places also may offer summer jobs, even for students.

Government jobs.
Some national governments hire many mathematicians;
you may have to be a citizen of the country.
These places also may offer summer jobs, even for students.

Other. Who wouldn't want to hire a smart mathematician?
To find job opportunities,
visit MathJobs. If there
is a particular university you are interested in that has no
announcement on MathJobs, try looking for employment opportunities on
the math department's website, or, if you know a professor there, ask
the professor whether they have any positions you could apply for.
The same job title can have different meanings at different institutions:
For instance, an "assistant professor" position may be tenuretrack,
or may be a temporary postdoctoral position.
At a few places, such as Princeton, "assistant professor" positions
are technically tenuretrack, 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,
and this was back when each application had to be sent by mail;
admittedly, the job market was very tight that year.)
An application for an academic position at a university or college
typically consists of the following:
 Cover letter.
This should mention
 what position(s) you are applying for;
 your current position and institution;
 the month in which you expect to complete your Ph.D.;
 your Ph.D. advisor's name;
 the general area of your research (e.g., arithmetic geometry);
 a sentence or two specifying your interests more precisely;
 the reason you are interested in the position under question (e.g., names of professors with matching interests);
 if you are attending the Joint Mathematics Meetings, mention this, and if you are giving a talk there, mention the specific time and place.

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.
Definitely include the following if they are relevant to you:
 your full name, professional address, webpage, email address, and phone number (if you are a U.S. citizen or permanent resident, mention that too since some jobs are restricted to U.S. people);
 your education history (degrees received, including the subject, university, and year),
 your employment history (for a C.V., normally you would include jobs and teaching assistantships, etc. only if they are related to your academic or professional career  for example, if you worked at an ice cream shop during high school, you don't include that);
 honors, prizes, and other awards;
 mentoring experience (list high school students, undergraduates, graduate students, and postdoctoral fellows who worked under your guidance);
 editorial experience (a list of journals for which you refereed, and a list of editorial positions you have held if any);
 conferences organized;
 committee service;
 other professional activities;
 publications (give full bibliographic info); and
 preprints (for each, say whether it is in preparation, submitted, or accepted).
You may also include (and it is probably a good idea if you are junior)
 lectures given (eventually, after many years, when the list becomes too long, you can list just the important ones under a heading "Selected invited lectures"); and
 conferences attended.

Teaching statement.
Your C.V. should have already listed the classes you taught,
but here you might describe in a little more detail what
the nature of your teaching experience has been
(e.g., how advanced were the classes, how many students, etc.)
Mention any other teaching or mentoring experience you have had.
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
and that you are not crazy.

Research statement.
The focus should be on your results and your ideas for future research.
If possible, the first paragraph should be understandable by
any mathematician; later it can get more technical.
It is OK to include some background,
but it should be in the service of putting your results in context 
you are not writing a textbook.
Also mention previous work done by others if it is related to yours,
or 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.
It is OK to mention theorems whose proofs you have not written up
formally yet, but do so only if you are very sure that you have a proof.

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.
See these suggestions
on how to request a letter of recommendation.
There are many more good suggestions on the
MGSA site
and in the article on pages 10211026 of the October 2006 Notices.
 Responding to offers.
 If a place you applied to asks you informally whether you are still interested, generally you should say yes if you have not ruled them yet, even if you are waiting to hear from other places you prefer. They may ask what other offers you have; it is up to you whether to reveal this, but in any case you can tell them that you will keep them informed of your status (and then do so).
 If you receive a verbal offer, wait for a written offer (email or mail is OK) before accepting it or declining other offers.
 If you know that you might want to defer an offer, e.g., to spend a semester or year at a research institute with a program in your area, then in most cases it is best to ask about this after receiving the written offer and before accepting it.
 After receiving a written offer, if there are other positions you applied for that you are sure you are no longer interested in, then withdraw your application to them.
 A written offer will usually come with a deadline by which you must respond. If you are still waiting to hear back from other positions, you can try asking for the deadline to be extended for a week or maybe a few weeks, though the extension might not be granted. Asking for an extension is especially reasonable if the deadline was early.
 For postdoctoral positions, salary and teaching load are typically standard, not negotiable. For tenuretrack positions, however, it may be possible to negotiate salary, teaching reduction for the first year, startup funds, housing assistance, tenure clock (after how many years do you come up for tenure  sometimes earlier is better, sometimes not), etc.