What advice would you give students applying for graduate school in mathematics?

I personally had the experience of applying to graduate school from a very small college that has not produced many math Ph.D.s (though oddly enough, I'm not the only 2002 BA graduate working at an R1 university in North America) not so far in the misty depths of time. Of course, I only have a couple of data points, but let me try to give my thoughts.

I think one thing is you've really got to get the basic stuff right: take the hardest classes available to you from the selections you have (and you should note in your letters that the student did this!) and ace them. Do well on the general and subject GREs. I assume you could have told them those.

I think the biggest non-obvious thing is that it's essential to seek out experiences outside your small program. If you're at a large school with its own graduate program, probably you have all the resources you'll need at your fingertips, but in a smaller program, I don't think you really have the ability to fully prepare someone for graduate school. Study abroad is good for this (I went to BSM in Budapest, but there are other programs). So are REU's (I did the one at LSU). On a smaller scale, you might be able to take more advanced courses at a nearby college (for example, students at Smith and Amherst can take graduate courses at UMass), or do a guest semester somewhere in the US (like at the Penn State MASS program).

These will, of course, be generally enriching experiences, but they also help by giving some real points of comparison. Graduate schools know what an A at BSM means (where they might not know it for your school); a professor who supervised you at an REU can speak with authority about having supervised many talented young people, and having some experience with which of them succeed in grad school.

Another possibility is working at summer mathematics program like PROMYS or Canada/USA Mathcamp (there are many other) though I think you should give some weight to activities like REU which are more likely to result in letters.

EDIT: I'll just note that what I've written about is still advice from the Web 1.0 world, but it's not so clear to me what Web 2.0 has changed about the admission process. I think it has made communicating with other people going through the process easier (for example, you can see in essentially real time which schools are sending out acceptances and rejections. This seems more likely to drive you insane than help though). I think for students at a smaller school this can be a boon (for example, the applicant profiles in this thread could be helpful for understanding where they might get in, though it's worth a reminder to take things with a grain of salt).

I think one thing I didn't know was that it's very reasonable to contact the Director of Graduate Studies at programs that interest you if you have questions. Don't be a pest about it (only email if you have a real question), but communicating with applicants is part of the job, though my DGS may not appreciate me telling the internet that (sorry, Tom).

The Web, of course, has also changed research about grad schools a lot. In theory, you can know a lot more about individual professors now than it would have been easy to figure out even 10, but especially 20 years ago. It's not super clear to me that this will help very much though. I generally feel like researching individual professors before starting grad school is a mug's game, since you're so likely to shift interests. I think it's much wiser to choose based on the program, and then worry about getting an advisor after a year or so of grad school when you know a bit more.

Ben Webster has already wrote a great answer about how to get into a good math grad school, so I will not repeat that answer here. I do, however, want to interpret your question more broadly (what advice would I give to seniors applying for grad school?) and give you some advice. In particular, most of the things that I am about to talk about were not so much of problems back in the 70s, I think.

No one tells you about the darker side of academia when you are an undergraduate student. But academia is really, really hard. Here are some cons of becoming an academic.

• the most serious problem by far is the fact that you do not have any choice on where you live. You just go to the best grad school that you got into, then after that, you apply to around 60 institutions for your postdoc job, and cross your fingers. Most schools give you two weeks to decide on your position, then off you go to some random city that you have never thought of. The same deal repeats for your second postdoc (if you are unlucky), or tenure-track. If you fail to obtain tenure, you might have to do another postdoc or tenure-track. So you are actively changing cities every 2-3 years for the next 10-15 years of your life. As an undergraduate student, this is not a serious problem, but when you have a significant other, and you are past 30, you really do want to settle down.

• academia is an extremely hierarchical society. Whether you get a job or not depends on your letter writers, one of which must be your advisor. As you obtain letters from people senior to you, it is often very difficult to express your opinions, as it is very easy to burn bridges. Academia is also very small; words travel fast. If you mess up in one place, chances are, everyone knows about it. Sometimes, it's not even your fault, but people can get very, very upset at you.

• the job market is very tight these days. The schools that I consider to be the top grad school are Harvard, Princeton, MIT, Stanford, Berkeley, Michigan and Chicago. These schools alone produce about 100 PhDs a year. I think that there are around 300 research postdocs available each year in the US (if you apply to every PhD producing institution, I think that you would send out around 80-90 applications; let's assume each school hires around 3 people, which I think is being generous). You can do the math.

These are only the universal issues that apply to practically every PhD student. Then there are issues that apply more strongly to some people than others (elitism displayed by some mathematicians, competition among peers, difficult advisor, your thesis problem being scooped, etc.)

So I advise that you need to be very, very sure that you really love math. I have been through many of these issues, and honestly, your love for the subject is the only thing that will keep you going. If you don't have the passion, you will regret your decision very quickly.

There are also perks of being a mathematician, of course. I do enjoy the flexible schedule (aside from teaching and committee work, research can be done any time, anywhere), being able to travel to conferences, and the job security that follows, if you make it to the ranks of a tenured professor.

This is too short for an answer. However, I do think it is worth an answer.

Tell your students to get on this site ! There are plenty of questions and answers about graduate school admission. If they don't find their questions answered, then ask their own questions to get useful advices.

Your students may have the same general questions as how to apply for admission. They may also have specific questions regarding their individual situations. The best advice is, get on an excellent, informative and reliable Q&A site and that's us!