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I’m a mechanical engineering student and I am about to apply for a PhD. I am certain about my desire to pursue advanced studies. However, I have serious doubts in what I want to pursue my studies on.

I have done research on structural health monitoring and power electronics. At some point I thought that I had interest in electrical engineering and after some research I got interested in mathematical physics and then in applied mathematics.

I started doing undergraduate research when I was a sophomore in the area of structural health monitoring. However my job was to write an algorithm to implement the methodology. I realize now that what interested me was not structural health monitoring, but to study the origin and derivation of the theoretical equations. I saw the profiles of various faculty in applied mathematics and I found some of their research interesting. Doubts about applying:

  • Although I have a GPA above 3.60 some programs require certain number of courses in mathematics. I only have the mathematics included in the mechanical engineering program and one course of elementary statistics. I have taken 5 courses in mathematics. However I don’t show strength because out of the 5 courses, in 2 of them I got B when I was a freshman. I think that the taking the elementary statistic course last semester was not a good idea. I should have taken a more advanced course.

  • I have not taking any advanced course in mathematics.

  • The application to the program requires the GRE, TOEFL and subject test in math. Some graduate programs say that the GRE math are not mandatory but not taking it puts you in disadvantage. I am not sure I will have time to prepare for it and as a result I don't know if I will do well.
  • I have research experience, but it is not in this area. In addition, the reference letters will be made by engineering professors and not math professors.
  • Since I am about to finish next semester, I don’t have much time to take more courses or do research in the area.

I looked at a few topics that I found interesting in applied mathematics but I haven’t done research in them. As a result I don’t know if I will get tired or if I will change my mind once I start on the program. If I were to apply for this program and I change my mind would it be easier to change even if it means going to other department?

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This depends heavily on which country, university and field you are in. For example, switching between mathematics and theoretical computer science can be easy in some circumstances. –  Raphael Jun 25 at 9:05

4 Answers 4

Unfortunately, no, it's not so easy to change PhD programs.

It isn't like changing your undergrad major, where you just have to fill out some forms and get your advisors to sign off. At the graduate level, each department does its own admissions. So if you are in a mechanical engineering program and want to switch to applied math, you have to submit an application to the applied math program just like anyone else. You'll be competing for a seat against every other applicant from around the world, most of whom will have completed a full undergrad mathematics major.

You might be able to improve your odds by getting to know faculty in the math program before you apply, perhaps discussing (or working on) research projects with someone there. However, keep in mind that you will also need to stay in good standing in the engineering program, and you won't have a lot of spare time to explore math opportunities.

Also, from your description, your math background sounds much too light for entry into a math grad program. Applied math can be more flexible than pure math, but having no advanced courses at all doesn't sound too good. In fact, I'd be concerned that without advanced coursework, you may not even have enough of a sense of the field to know that it's really what you want to do. Mathematics courses have a sharp change in flavor and difficulty as you transition from lower-level computational courses (calculus, differential equations, etc) to upper-level theory courses (real analysis, abstract algebra, topology). Entering a PhD program having only the former kind of courses would be roughly analogous to being a go-kart enthusiast racing in the Indianapolis 500. And it won't be easy to pick up extra math courses while in an engineering program; grad program requirements rarely allow for free electives.

Perhaps something that might better fit your situation is a post-baccalaureate ("post-bac") program. These are one-year programs designed for students who have completed a bachelors and want to attend grad school in math, but need further preparation. Such programs are offered by relatively few institutions, but I believe some of them are quite well-known in the community and have very good placement rates.

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+1. I was just in the middle of writing almost exactly the same thing. One way of summarizing the central point of your second paragraph is that in graduate school you don't have a major, you have a department. –  Mark Meckes Jun 25 at 6:45
    
What is an example of one of the well-known post-bac programs you mention? –  Mike Miller Jun 25 at 7:46
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@MikeMiller for women and mathematics math.smith.edu/center/postbac.php –  BSteinhurst Jun 25 at 21:14
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@emory: That really depends on the field and administrative circumstances. In some fields, like math, your funding is not tied to your advisor (or you may not even have funding), and changing advisor's within a department is trivial. On the other hand, changing research focus while working with the same advisor may be easy, but actually switching departments would be more complicated. –  Mark Meckes Jun 26 at 4:54
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@emory: In fact, most math PhD students spend 1-2 years without having a thesis advisor. Their eventual advisor has little or nothing to do with their admission to the program, and many advisors -- even very eminent ones -- either have no funding of their own or cannot use that funding to support students. Finally, in many departments, students must pass most of their exams before they get an advisor, and many potential advisors have a policy of not taking on students even informally until they pass their exams. –  Pete L. Clark Jun 26 at 6:22

My advice is the same as Nate Eldredge's, but let me make a few points.

I am certain about my desire to pursue advanced studies. However, I have serious doubts in what I want to pursue my studies on.

To be very honest, for me the second sentence largely nullifies the first. Wanting to pursue a PhD in the abstract is a poor idea. In order to get any satisfaction out of a PhD program, you need to be pursuing a PhD in X because your love for X is so strong that you want to have a career doing X. If you think it might be X,Y or Z, I worry that you don't have a deep enough commitment to any of the three to make it worth the time, effort and career uncertainty of today's very harsh job market.

Perhaps by expressing "serious doubts" you are really explaining your desire to transition from your undergraduate field to mathematical physics / applied mathematics (henceforth I'll just say "math": that's where you would apply). I agree with Nate Eldredge that your coursework in mathematics is so minimal that you probably haven't been exposed to the flavor of graduate level mathematics....or at least, you haven't demonstrated that exposure in a way that would be convincing to an admissions committee. Further honesty: your self-described profile would make you a very likely decline from the math PhD program at UGA (about the 50th best in the country), except perhaps if you took the math subject GRE and did well on it. It would be a little surprising if you did well on that exam given your description of coursework...but of course it's possible, especially if you are very mathematically talented. It could be useful information for you to know how you might do on this exam, but unfortunately this exam is notorious for having had the same crappy study materials for the last 20 years (or more?). I don't know of any reliable way to gauge your score.

There are a lot more than 50 math PhD programs in the US, and I can't speak to your admit-worthiness for all of them, but honestly I would think twice about scraping the bottom of this particular barrel. The job market is so tight right now that degree pedigree is something to keep your eye on.

In conclusion, I come back to agreeing with Nate Eldredge: you are not currently a good PhD candidate in math, and math faculty like us would worry about your lack of exposure. Of course this does not mean that you are locked out of further study of mathematics: that never happens to anyone. But if you think you are interested, find out by taking more math courses, in a non-degree program if need be. I should say though that your stated background of five math courses required by an engineering degree -- at least two taken as a freshman -- is less than halfway to a graduate-preparatory math major. In your case, after enrolling in a non-degree program for a semester or a year, if you really discover that this is what you want to do, you might actually want to explore a second bachelor's program. Either that or really work hard to develop a relationship with a department so that you can essentially do this over the course of 2-3 years with a lot of faculty contact.

Good luck.

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+1 for "Wanting to pursue a PhD in the abstract is a poor idea." In fact, wanting to pursue a PhD in the abstract comes dangerously close to the stereotype of staying in school for the sole purpose of avoiding the need to decide what else to do with your life. –  Mark Meckes Jun 25 at 8:56

A PhD is not like an undergraduate degree where you can do a bit of this, a bit of that, add up your scores and get your final grade. It's about doing research in a fairly narrow area, leading to writing a focused thesis on a particular topic. Spending, say, a year doing electrical engineering would normally give you almost nothing that would be useful towards a PhD in mathematics, and vice-versa. As such, transfering from one department to another is, essentially, starting a new PhD and would likely be treated as such.

It sounds like you don't know much about what a PhD involves and you should definitely find out before committing several years of your life doing one! I'm also a little confused by your statement that you're certain you want to commit several years of your life to studying a fairly narrow topic but you're not even sure what broad area that topic will come from. It seems that you're much more excited by the idea of doing a PhD than about any of the areas that you might research, yet doing the PhD requires being in one of those areas that you seem to find less exciting than the PhD itself.

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I typed my answer before I saw yours. Obviously I agree with it: +1. –  Pete L. Clark Jun 25 at 8:18

Much depends on the specific university and department arrangement.

However, moving departments at the PhD level is usually quite difficult because it involves changes in your financial package. Transferring from physics to mathematics, for example, would mean your funding is coming from a different source. It also means that physics is losing a student, which can be bad for the department. Things are even more complicated if your funding is tied to a lab or an individual faculty member.

Many places consider transfers alongside new applicants (meaning, you are, in effect, applying to graduate school for the second time). You would have a slight advantage of knowing your professors, and perhaps of having an advocate for your candidacy, but you still have to write essays and get recommendations along with everyone else. I would not count on a transfer as a plan. You could just as well not get in, losing support from your home department in the meantime.

For these reasons, I often recommend students take some time between graduation and graduate school. Get a job at a lab or with a firm, take some courses, travel. Graduate school is like marriage--a long-term commitment. It is not a good place to explore or experiment with careers. There is no rush to make the decision, but once you do, make sure you are committed to your department and to your chosen discipline.

Master's programs are a different ballgame, since they do not normally fund students. It is probably easier to switch out of a Master's program. But once again, you would probably have to apply anew.

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"For these reasons, I recommend my students take some time between graduation and graduate school." You recommend this for students contemplating a PhD program in math or physics? I am not sure this is wrong (or even that it is meaningful to call such an opinion "wrong"), but it is at odds with the standard practice in this area. –  Pete L. Clark Jun 25 at 7:04
    
Moreover, I will say that students who try to get back into mathematical studies after several years away tend to do so at a disadvantage compared to others who go straight through: the type of coursework that someone does at the end of a PhD-preparatory undergraduate math major is much more like what will be done in graduate school and much less like either what has been done before or what one is likely to encounter in most non-academic walks of life. I wonder if you could speak to these issues in your answer. –  Pete L. Clark Jun 25 at 7:06
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Taking a job at a firm or travelling are what I meant when I referred to non-academic walks of life. I admit to confusion about your advice to take time between graduation and grad school by doing coursework: sandwiched in between other non-academic things, I guessed that you meant taking courses unrelated to the intended graduate study. If you mean instead that the student should consider doing post-bac work: yes, Nate Eldredge has recommended that. My answer would be the same as his. –  Pete L. Clark Jun 25 at 7:40
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"Of course going through without a break offers some advantage, in any discipline." It is my understanding that there are some disciplines in which it is virtually mandatory to take time off between undergraduate and graduate school: e.g. social work and business explicitly encourage this. Mathematics is a different field. It may sound small-minded to say that real world experience tends not to be directly helpful for graduate study in mathematics...but that has been my real (academic) world experience. Most students who take time off do not come back. –  Pete L. Clark Jun 25 at 7:43
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@PeteL.Clark Most students who take time off do not come back — This is compelling evidence that for most students, taking time off before grad school is the right strategy! –  JeffE Jun 26 at 0:49

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