The following are just some random snippets I extracted while reading “A Mathematician’s Survival Guide: Graduate School and Early Career Development” by Steven Krantz. I don’t necessarily endorse any of this advice in particular, I just found these snippets to be interesting things to remember and think about.
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It is of the essence that you work in a subject area, and on a thesis problem, that you like and can develop some enthusiasm for. Just as an athlete will say, “No pain, no gain.”, so a Ph.D. student in mathematics might say, “No passion, no thesis.” Just so, you also want a thesis advisor who can light a fire under you and can find a problem that will absorb you. You want a mentor who will inspire you to strive for excellence and achievement. You also want a thesis advisor with whom you can work comfortably and have a congenial relationship.
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Be courteous and friendly to the staff. They are the people who hold the department together and are there to help you with your work. If they do something especially nice for you, bring them cookies or flowers.
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I have covered the chapter and verse of how to teach, and in particular how to teach recitation sections, in the book KRA1. I shall not repeat those insights here. Let me just conclude by noting that you must take your teaching duties seriously. Learn your students’ names. Show them that you care. Make yourself available outside of class. Be fair and even-handed. Your department depends on you to do a good job, the math department’s reputation around campus hinges on you and the other math teachers.
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Am I Supposed to Work All of the Time? Definitely not. Fred Almgren, my friend and faculty mentor in graduate school, liked to say that graduate students should work four hours per day. What they did beyond that was their own business. Now one should bear in mind here that Fred had extraordinary powers of concentration. Four hours of work for Fred was like ten hours of work for anyone else.
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…problems on the qual, then you had better get seven or eight of them almost entirely correct. If some of the questions ask you to state theorems or definitions, then you had better get them letter perfect-with proper English and all the quantifiers in the right places. What you are learning is a discipline and your work had better manifest that discipline.
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As I’ve stated previously, a qualifying exam is not like a calculus test. You will not pass a qualifying exam on partial credit alone. The examiners are trying to determine whether you are qualified to do thesis work. Can you see to the heart of a problem? Can you write a proof well? Can you recognize correct mathematics and incorrect mathematics? Can you think critically? As a result, what the examiners want to see on your qual is a substantial number of questions answered substantially correctly.
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Well, you don’t want to flunk too many quals and you don’t want to flunk them too often. Every program will give you at least two tries, and different people mature (intellectually) at different rates. The quals are not like the thesis. Qualifying exams are just a basic learning situation, one at which you have excelled all of your life. If you apply yourself and follow the advice given here (and of course follow the advice of your advisor), then you will certainly get through the quals.
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…the questions is terrific training. Learning to answer them is an even better regimen. And you can work out the answers together. Learning to talk about mathematics is an essential part of any graduate education.
- So how do you study for an exam that contains such questions? Typically, you are not going to find questions like these in books. And, even if you do, you are not going to find the answers laid out for you. Let me put it this way: When you study for a qual, it is not enough to learn just the statements of the theorems and the proofs. Of course you must do at least that much. And at some schools this basic effort may be sufficient for a low pass. But the new thing that you must learn to do in graduate school is to ask yourself questions. Turn the ideas over in your mind and ask,
- Why is the theorem stated this way?
- Why is this hypothesis really needed?
- What happens if we change the conclusion from this to that?
- What would be a counterexample?
- Why does the proof go like this?
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I will conclude this section by enunciating a very important principle (which will be repeated often in this text) of getting an advanced education. You are no longer learning calculus or another trivial subject where it is sufficient to read the text and do the homework. You are now doing the toughest thing you will ever have done in your life. It is essential that you talk to people-all the time. In this way you can orient yourself, keep to your course, be sure you are doing the right thing, and have a constant reality check. It is also an important part of being a mathematician to be able to communicate - not just technical mathematics but also information about mathematics, about teaching, and about the profession. You are now not simply learning mathematics - you are learning to create it. So my advice is to talk to your fellow students and to the faculty (and to the staff) about everything. Eat lunch with a group, socialize, talk to your office mates. This is your new life.
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One thing that you begin to realize while you are a graduate student is that learning does not have to be a formalized process. You do not need to take a course, with a teacher, homework, and a grade, in order to learn a new subject. By the time you reach an established level in the academic world, you probably will not have the patience to sit through courses; instead, you will learn things entirely on your own. An intermediary step to that lofty position is to develop the habit of auditing courses. One of the things we were taught right away in the Princeton graduate program is that we should think of the courses just as we think of the books in the library: these are resources that you can drop in on, and drop out of. You don’t need to register. Just access them as your interests and your studies dictate.
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Always remember that the qualifying exams are not the point of graduate school. They are just a step along the way. The main thing is to write a good thesis. So your short-term goal, at the beginning of your graduate career, is just to get through those quals. The quals are a zero-one game. Once you have passed them, then you need never look back. It’s time to write the thesis.
- Make a point of getting to know some of your classmates (your peers) and also some of the graduate students who are ahead of you in the program. The latter bunch will be full of a lot of gossip and a lot of baloney, but they also have passed the quals and they are familiar with how the program works. They know which classes to take, and the various hoops that a graduate student must jump through. Pick their brains. They can tell you which quals are hard and which are easy. Who writes the quals and who grades them. How quickly you are expected to get through the qualifying exams. They will know who the good instructors are, who gives good courses, and who the good thesis advisors are. This is vital information that you must know and understand.
- (Ask them) where the graduate students hang out, what is expected of you. Where the coffee pot is. Do the same with the Department Chairperson.
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(If) this is your first day in graduate school, I admonish you to take charge of the situation. Figure out where the math building is, go there, and introduce yourself to the Chairperson’s secretary and the Graduate secretary. If the Graduate Chairperson is around, shake hands and introduce yourself.
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As your program develops, keep in touch with the Graduate Director and with your thesis advisor to make sure you are making good progress. The rest should take care of itself.
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It is my fervent belief, well-supported by experience, that the main reason that people often fail at tasks or programs that they set for themselves is that they never figure out what it was that they were supposed to be doing. The Ph.D. program in mathematics is a multistep, fairly complex process. There are many junctures at which one could lose track and not get the right mentoring or advice. The purpose of this book is to provide some objective reference material, presented in an accessible but authoritative tone, to aid in the graduate education process.
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Of course a collaboration is like a marriage and you must manage it with the same delicacy. Some very fine collaborations have fallen by the wayside because of priority disputes or personal differences; this is just deplorable. Read [KRA2] and also [KRA6] to find out more about how mathematical collaborations function.
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How do you establish such an international reputation? Well, you must publish, and in good journals. Talk to people. Go to conferences. Give talks. Share your ideas. Collaborate with people. The strategy is to let people know who you are and what you have to offer. You want the established people - around the country and around the world - to think of you as an “up-and-coming person”, one whom they are happy to assess and praise. You want to be a person whose papers are read and quoted.
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Of course I don’t need to tell you that you must be on top of your mathematics. The people interviewing you want to know what kind of mathematician you are. Be prepared to discuss your thesis and what your current research interests are. If you can get involved in a serious mathematical “chalk talk” with people -standing at the blackboard and doing mathematics - then you are sure to make a good impression.
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What really makes a terrific impression in your job application dossier-at least to a research institution-is getting some letters from faculty outside of your university. I often invite colloquium speakers who would be of interest to my graduate students and arrange for the students to spend some time with the guests. This frequently results in the outside faculty member becoming familiar with my student’s work, perhaps even reading the thesis. Then this person can write a really nice letter on behalf of my student. More than one of my Ph.D. students has written a paper with a mathematician at another university - even before graduation.
- For the most part, you must go through the formality of submitting job applications. This includes
- Completing an AMS Cover Sheet (available from the American Mathematical Society web site or in issues of the Notices of the AMS),
- Putting together a Curriculum Vitae (see [KRA2] for some advice on how to write your vita),
- Getting letters of recommendation (usually three),
- Writing a Teaching Statement, and
- Writing a Research Statement.
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(To which) journal should you submit? The book [KRA2] contains extensive detail on how to write up a paper and on the process of submitting the paper to a journal.
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You may have been a child prodigy, but now you are just another mathematician. Your progress, achievements, and contributions are probably like everyone else’s. On the other hand, mathematics is one of the finest and most erudite achievements of the human mind. The scholarly standard in the mathematics profession is one of the highest in the academic world. It is something to be proud of. And you are part of it.
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After you have graduated and you have been in the profession for a while, it will appear that
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Everyone else is getting invited to all of the big conferences;
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Everyone else has lots of research grants;
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Everyone else is winning Sloan Fellowships;
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Everyone else is getting invited to speak at the International Congress of Mathematicians;
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Everyone else is getting the plum jobs.
- …at that time was for the advisors to just shuffle us off to the plum jobs - the Moore Instructorships at MIT, the Dickson Instructorships at Chicago, the Peirce Instructorships at Harvard. It was a halcyon time…
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Other prestigious appointments:
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Elected to the National Academy of Sciences;
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A Chair Professorship at Harvard;
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Winning the Wolf Prize;
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Getting the Steele Prize.
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(Experience) failure and disappointment every day, and try to find ways to surmount it. For the sake of success in mathematical research, it is much more important to be able to cope with the frustration inherent in the process than it is to be “quick” or “brilliant”. To be sure, the top people are often successful, lucky, ….
- The only way to get the writing process going is to begin. Start small. Generate a broad outline of your thesis. The first pass can be a very vague adumbration, with just the key topics laid out. Then develop the outline by gradually adding detail. Show it to your thesis advisor just to make sure that you are headed in the right direction.
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Now create a very detailed outline. This draft should actually list every definition, every lemma, every theorem, and every example. Don’t write them all out. Just indicate each one with a couple of words: “the covering lemma” or “the spectral sequence argument” or “the completeness axiom”.
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The more you do mathematics, the more you will treasure concrete examples. I would say that most of my best work is based on just a few examples that I return to over and over again. Always remember that we learn inductively (going from the specific to the general) rather than deductively (going from the general to the specific). The deductive mode is highly appropriate for recording mathematics, but it does not work for discovering mathematics. You discover, and create, mathematics by starting small, by doing little calculations, and then working up to more substantial and meaningful calculations. The aggregate of many calculations can become an insight and that insight might turn into a conjecture. Even more effort might transmogrify that conjecture into a theorem. It is a fantastic process.
- Learn good work habits.
- When a calculation finally works out, write it up carefully, number the pages, date it, and put your name on it. File it away in an organized manner so that you can find it again.
- Keep a daily journal. Record in it what you have tried, what works, and what does not.
- Be aware of the fact that, when you do a calculation or make a discovery, it will seem as plain as day and something you will never forget. Sadly, you will. And you will forget it most surely exactly when you need it. So write it down.
- Often things that you tried and did not quite work out are just as valuable as things that are beautiful and glow in the dark. Keep a record of everything you try. This will be valuable both for psychological support and also as an archive of your efforts.
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Let me assure you that, in working on your thesis, you will not get better every day. Some days you will seem to learn something new and get closer to the goal. Other days you will have to tear up what you did the day before and try again. There will be days when you seem to be blocked and others when it appears as though you are on a fool’s errand. Don’t let it get you down. This is the life of a mathematician. Your entire career is going to be like this. At least now you have senior mentors to guide you.
- (On how to finish a thesis) The short answer to the question is: You work on your thesis problem by sitting in a quiet place and calculating. You try things and then modify them and jiggle them and then try them again. Fill dozens and dozens of pages with your speculations and trials and scribblings and conjectures, then throw them away and start again. During this process, you are constantly talking to people, going to seminars, writing e-mails, and asking questions. You immerse yourself in the problem and swim around in it until you find something that floats - something to latch onto. Gradually, you develop that handhold into a thesis.
- This is neither a trivial nor a naive question. The answer is not, “Just sit down and do it.” I had a friend in college who did all his homework assignments, and his takehome exams as well, by poring through every book in the library until he found something that looked like a solution to the exercise on which he was working. This worked fairly well. He received reasonably decent grades but he did not really learn much of anything, because he never put his mettle to the test. You develop your brain by banging it against things, by stretching it, and by challenging it. This means that you must do the work yourself.
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If you are stuck on what you are doing, or if you are discouraged by what you are doing, or feel that you are not making any progress, then please say sol! Your professor is not a mindreader.
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When an experienced, senior mathematician gives you a thesis problem, then s/he is doing you a tremendous favor. Your advisor is, in effect, saying, “Here is something worth doing and it is at your level. It is a doable problem and you will get a publishable paper out of it. People are interested in this topic and you will begin to make your reputation by solving this problem. Moreover, working on this problem will lead you to other worthwhile things later on.” This is something that very few young mathematicians can do for themselves. You are not well enough read and not sufficiently networked in the mathematical community to know which problems are interesting and which are not. You certainly do not know which problems are tractable and which are not, and you do not know which problems fit your abilities. When things do not work out, you do not know what else to try, or how to adjust the problem to make it more feasible.
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(On Krantz’s advisor) The thesis problems that he gave me were completely unfamiliar and they were all quite difficult. But they were good, they were the kinds of problems that you could jump in and start working on right away, and they had connections to many other things. In other words, they were ideal thesis problems.
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A more typical scenario is for the advisor to say, “Why don’t you read this paper of mine and see whether it gives you any ideas?” or “Why don’t you read this paper and we’ll talk about it?” or “Why don’t you proofread the chapters of my new book? It’s a good way to learn the subject and there are a lot of good problems posed in there.” This is fine, and you should do what you are told. Again, don’t be bashful. Talk about your reading with everyone - with your fellow students, with other faculty, in seminars, and of course with your advisor. It is definitely not a zero-one game. It is not as though you can only speak when you have a theorem to show. You can speak any time you like, for almost any reason; and you should.
- When I advise undergraduates applying to graduate school in mathematics, I always tell them the same thing: “Go to the best school you can get into, find the hottest, smartest professor around, sign up to work with that professor, and do anything he or she tells you to do.” That is what I did, and it worked like a charm. Of course this is not necessarily the best advice for everyone. Some students do not want to go to an extremely competitive program…
