Matt Might, Assistant Professor at the University of Utah, writes, “Every fall, I explain to a fresh batch of Ph.D. students what a Ph.D. is. It’s hard to describe it in words. So, I use pictures.”

Click below to see Professor Might’s “Illustrated guide to a Ph.D.”

http://matt.might.net/articles/phd-school-in-pictures/

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Will and others,

Interesting illustration of what a Ph.D. is! And it is a good way to describe in rough but understandable terms the differences between Bachelor’s, Master’s, and Doctoral degrees.

However, we should be careful to note that many people with PhD’s do have interests and even good knowledge of many subjects well outside their research work. A popular stereotype of those with PhD’s is that they know very little of anything outside their research area. Of course, some do fit that image, even within their own subject taken broadly (for example, some with math PhD’s have little knowledge of even math outside a narrow research area). But many do not.

There is a link on this page called “10 Reasons Ph.D. Students Fail.” Some of these do not seem to apply to UK in mathematics where I had tried a Ph.D. You can never learn too much mathematics, especially these days when so many areas of mathematics are finding startling connections between them. It is always possible that learning something seems a waste of time because it does not seem to have any connections with your research interests, but sometimes you never know. However, I do agree that one would not want to spend too much time chasing after unrelated or unnecessary knowledge, especially not in graduate school and especially not before passing preliminary exams because you have to pass those to proceed! Speaking of preliminary exams, I find it absolutely shocking that this document mentions nothing about them, yet at least 99% of UK math students who fail to get a doctorate in math failed to do so because they could not pass the preliminary exams by the deadline (this does not include those students who had decided to quit early with just a Master’s, but I have no idea how many do quit simply because of the prelims or other reasons).

I agree that grades are not terribly important as long as you keep the minimum GPA. But here is a catch: If you don’t focus on your coursework in mathematics, you are likely to fail the prelim exams because they cover that coursework! If you fall behind in studying for a class, you will soon find yourself miles behind with a huge load of catch-up studying to do. And all that catch-up studying is absolutely necessary to give yourself any chance of understanding what the course is covering now! For instance, I cannot read Chapter 5 of most graduate math textbooks without understanding Chapters 1-4. I might not have to understanding everything in those previous chapters, but you can bet I will have to understand a decent amount of the material within those chapters.

Don’t think that you can “catch up” on tons of studying after the semester ends and before the prelims because you don’t have much time to study. The only exception is if you are exceptionally brilliant for a PhD student in mathematics.

And following the advice

“During the first two years, students need to find an advisor, pick a research area, read a lot of papers and try small, exploratory research projects. Spending too much time on coursework distracts from these objectives”

does not work very well for most mathematics students beginning graduate school because not many of them have any idea of what area of mathematics they wish to specialize in. At least it does not seem to work well at UK because the professors there don’t want to bother with you if your project isn’t required for a class and is completely unclear from the start that it will lead to original research.

Another reason that this does not work is that you cannot take all your prelim exams in the same area of mathematics. Even if you love (abstract) algebra with a passion (say, so much that you officially get married to algebra), you will have to take prelim exams outside algebra, even if some of those exams never turn out to be useful for your research. And, as I had mentioned above, you never know in advance if some area of mathematics outside algebra will be useful in your research. And this is assuming that what you pick at the offset will be related to your research for the Ph.D., which may or may not be the case.

And few of the professors are interested in becoming your advisor this early because your interests may change and also because of the risk of failing to pass the prelim exams in time. Some of them might appreciate your interest and be willing to take you on eventually but usually even those professors will not do so that early. And, because of the comments I had mentioned above, most would be concerned if you concentrate too much on such outside-of-class projects, you will not be able to study enough to pass prelim exams in time.

As for avoiding perfection in writing, this is often followed in graduate school and often followed by professors in practice. The author is correct in that we cannot polish a paper forever, but a good research paper needs more than two or three drafts to get it in very good shape. So we cannot demand perfection, but taking an extreme position on the other end (“heck, I can’t be perfect in my writing, so let me just write something, look at it a couple of times, and hope it’s okay”) results in many papers being published that are extremely difficult and sometimes painful to read because the mathematician clearly cares far more about quantity of papers and treats each one as an emergency project. I believe it is far better to publish several very well written and very insightful papers than to publish 2000 papers that clearly reflect rushed research and writing. However, many big research schools in mathematics think the exact opposite. In short, we cannot demand perfection in writing, but that does not give us an excuse to write sloppily.

Finally, I will close with some comments on the following advice from this article:

“It does not matter at all what you get your Ph.D. in.”

“All that matters is that you get one.”

“It’s the training that counts–not the topic.”

In mathematics, this is true to some extent, but then again, depending on your career goals, what you get your PhD in does matter. And it is important to note that switching research areas in mathematics, especially switching to an area that is far from your current area, is extremely difficult to do. Some mathematicians are able to do that without “too much” trouble (whatever that means), but that is not the norm.

So though this advice is true, that previous statement should be taken into account. It would be quite a rough road later on if you decide you want a mathematical career that demands research specialty in PDE’s (especially arcane knowledge of PDE’s that very few mathematicians outside PDE’s have) when your current research work does not require you to know much about PDE’s.

Of course, this advice might very well apply to PhD students in general, but I cannot tell for sure because I don’t know much about getting PhD’s outside mathematics. Then again mathematics is a unique subject with many traits not found in other subjects.

Jonathan Groves