For a good and broad view of the problem, buried in D. Knuth's The TeXBook is

"The traditional way is to put off all creative aspects until the last part of graduate school. For seventeen or more years, a student is taught examsmanship, then suddenly after passing enough exams in graduate school he's told to do something original."

Yes, here Knuth identifies a significant challenge.

Compared with the OP, here's a very different and much more specific approach that clearly makes a lot of sense and that worked well for me:

First, note carefully that for some work that can be called research the usual, nearly universal criteria are that the work be "new, correct, and significant". Below, keep these three in mind.

Second, get a major in math, at least a good undergraduate major in pure and applied math and hopefully enough more in pure and applied math for roughly a Master's in math. Even if you don't care about the Master's degree, I do very much recommend getting the math for a Master's degree.

Why pure math? The pure math gives you the crucial, central, foundational tools of math, that is, many crucial prerequisites and, broadly, the ability to state and prove theorems. E.g., you will learn how to write math, and that alone will start to put you ahead, even of some high end professors.

What pure math? For your research likely mostly you will use the part of math called analysis but in your studies for more you will also want at least the basics of abstract algebra and maybe differential geometry, combinatorics, and maybe even some in foundations. In addition, if you have some reason to believe you can get some value from algebraic topology or geometry, then, sure study those.

Why applied math? Likely applied math will be closer to the math you will use for your research. What applied math? Sure, e.g., statistics, numerical linear algebra, ordinary differential equations, more in numerical techniques, optimization, stochastic processes, etc.

Third, get your Ph.D. in some field of engineering -- computer science, electronic engineering, mechanical engineering, civil engineering, operations research, statistics, etc.

Three biggie points:

(1) In science and engineering, by far the most highly respected research is that which mathematizes the field. Good work here can help meet the criterion of "significant".

(2) Work in math, well supported with theorems and proofs, is much more difficult to criticize than work that is mostly just experimental or empirical. Good work here can be help meet the criterion of "correct".

(3) The standard and severe weakness of the backgrounds of researchers in most of science and in engineering is way too little in math. Thus, there are a lot of good research problems they can't address. So, your good work here can be help meet the criterion of "new".

So, with your background in math, on (1)-(3) you will have at least a good -- maybe even an overwhelmingly strong -- comparative, competitive advantage.

Another point if you care: Unless your family wants to donate $10+ million or so, it is just super tough to get into an Ivy League university. But getting in as a grad student is much easier -- e.g., I got accepted to Cornell, Brown, and Princeton.

So, you should intend that your research be essentially math for that field of engineering. Usually you will aim to use your math tools to solve a relatively practical problem in that field of engineering, but you might use your math to add to the basic theory of that field; for some wild guesses, you might do something in the theory of predators and prey in environmental engineering; maybe you would have been the one who did Kalman filtering in electronic engineering; maybe in mechanical engineering and continuum mechanics you will make some nice theoretical contribution to materials science.

Why engineering instead of pure math or physical science? (1) Engineering has no end of practical problems -- say, from outside of, and neglected by, academics -- to be solved. So, if you pick, attack, and solve a problem important in practice, then there is a good chance your work will meet the criterion of "new", since the work is mathematical, "correct", and since the problem was important in practice, "significant". (2) In pure math and physical science, the range of candidate problems is much more narrow, e.g., in physics you can try to say what dark energy is -- lots of luck doing that.

So, right, for a research problem in some field of engineering, maybe pick a practical problem that is considered important and that you found someplace, maybe outside academics, maybe on a job, maybe a real job or maybe just a summer job or an internship. I did that: I picked a problem I found at FedEx.

Then, it will be quite good for you to have the problem in mind when go for your Ph.D. I had the problem and a good, first-cut, intuitive solution (worked out on an airplane flight) before I entered my Ph.D. program. In my first year, I took some advanced, relatively pure, not often taught, graduate math coursework that gave me good math prerequisites to let me convert my intuitive solution a solid math solution. So, in my first summer, in six weeks, independently, alone in the library, I worked out the math, with theorems and proofs, and walked out with a 50 page manuscript that was the original research for my Ph.D. dissertation. I recommend doing such a thing.

Getting into research early is commonly considered good advice: E.g., IIRC, the Princeton math department has said on their Web site that a student should have some research underway in their first year. Even, better, have the core research done before the second year -- which is what I did and, I believe, a strong advantage in getting the Ph.D.

The math gave me another advantage: In a course, a problem was apparent -- a tricky, deep question about the Kuhn-Tucker conditions. There was no answer in the course, and I could find no answer in the library. So, I attacked the problem -- the key was some pure math I had -- and found a surprisingly nice solution, in two weeks. I wrote up my solution and got credit for a reading course. But the work was publishable -- presto, bingo, at that university the criteria for a Ph.D. dissertation was that the work be "an original contribution to knowledge worthy of publication". Well, the best way to show that some work is "worthy of publication" is to submit it for publication and have it accepted. I did that. So, technically that work was enough for my Ph.D. dissertation, a second one.

For that problem in the Kuhn-Tucker conditions and for my dissertation research, I never had any real faculty direction. I recommend: Don't wait for the faculty to provide a good problem or direction. Instead, on your own as much as you can, at least if it is easy for you, and it was for me, pick a good problem, do the research, get the work ready for publication, and, hopefully, publish it. For a graduate student to have, early on, from largely independent effort, some work worthy of publication makes essentially everything else in the Ph.D. program and the start of a career much easier and better.

Okay, how to do the research? Well, for me, the core, hard work of the research was a little more involved but, really, not much more difficult than the more difficult exercises in standard, advanced pure math texts.

The difference was, for research, in part need to keep in mind some view from higher up, say, 50,000 feet down to 1000 feet and don't always be crawling around on the ground with the lowest level details (which is common and usually effective enough in solving exercises).

Next, guess: To find and prove a new result, first have to guess it. Sure, make educated guesses based on your solid background but also work just intuitively. So build intuitive models and, as you learn more, revise the models to make them more accurate.

E.g., during the work, is A true? Well, it doesn't seem wrong right away intuitively. But, if A is true, then, hmm, B is true. Could B be true? At least, first-cut, intuitively, naw, not a chance (this may be wrong, but let that happen for now). So, likely A is not true.

You can do a lot of this in your head without writing anything. And, even if slowly, you will learn to do at least some derivations in your head.

Now, for C, intuitively it looks true. So, try to prove C. Gee, the proof doesn't work. Then observe: The proof doesn't make good use of all the hypotheses of C; so, you've been trying to prove something more general than C and likely not true. Bummer. So look again at the hypotheses of C and try to see how they are essential and how to exploit them.

So, continue in this way, maintaining a good view from above the ground level, with lots of intuition and guessing and trying to prove some little things.

When you get a proof of a result that looks good, then write it up, carefully, cleanly, put a date and title on the first page, put a staple in the UL corner of the sheets, and toss it on a stack, continue on, maybe building on what you have.

There is also Polya, How to Solve It.

From A. Wiles, the guy who solved Fermat's last theorem and just won the Abel Prize, is

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ..."

> However, I think you could also view it from the perspective of it often being easy to make a novel contribution in a field when you possess skills from a completely different field (i.e. interdisciplinary research).

A very old story is that field crossing is a powerful aid to innovation.

> Of course you could argue whether mathematics should be treated differently because it is in some sense more fundamental.

Yup, especially for the most respected research, mathematizing a field.

Well, that's debatable :) I can think of plenty of recent systems conferences where the best paper had nothing to do with mathematizing the field.

Yes, I work in systems. No, don't ask me what I system I'm talking about. I'm certainly not allowed to tell you, of course, and anyway there have been multiple such systems throughout my life.

So. Many. Race conditions.

Remember, the value of a PHD is maintained by rejecting people. If you demonstrate your 'in the club' then you can often just stable together a few papers and call it a day.

I've already done a pure math (+business) undergrad from a mid-tier university and have been working as a product manager at a financial services company for 3yrs post undergrad. I've been thinking of getting a PhD in an engineering sub-discipline, but not sure how to go about doing it especially since I have no research experience.

Do I:

(i) spend 1 year doing research with a professor nearby then apply (I live close to a major research university)

(ii) apply now and hopefully my work experience can cover for research experience

(iii) Rethink the whole PhD thing and get a professional degree like an MBA (did well on GMAT)

(iv) Just continue to learn on the side and try to use my skills for opportunities

-----

Also two more questions if you don't mind:

How did you narrow your interests before applying for a PhD?

Is it possible to get a masters level pure math education through self study? I've taken until PhD level measure theory.

If you know the discipline you want to study, email someone like the Director of Graduate Studies in the department in question. If you don't know the discipline, email someone in either the Graduate School or the College of Engineering, or both, depending on how you school's organized.

It would appear that you might pursue something in "financial engineering".

Since you are in the financial industry, you might ask around and grow a network: E.g., ask person A; maybe can get them to suggest person B; tell person B that person A recommended them and ask person B; etc. I.e., apply social networking techniques. E.g., try to get to some people who can outline some of what James Simons did. If you can, talk to Simons. Similarly for various quants and people who create algorithms and code for automatic trading. Look into the programs in financial engineering at Princeton (E. Cinlar), NYU (M. Avellaneda), Columbia (I. Karatzas), and CMU (S. Shreve).

If you want to get into some other field of engineering, then gather some information, meet some people and get some input, and start to pick some fields or one field.

> (i) spend 1 year doing research with a professor nearby then apply (I live close to a major research university)

Maybe. If you can find a good situation and like it, maybe, sure.

I have been suggesting that you start with your own problem and make some progress on it, maybe with a little advice (once I got just three words), and maybe then consider a professor as a official or unofficial mentor or dissertation adviser.

> (ii) apply now and hopefully my work experience can cover for research experience

For a graduate program, in most fields, you are not expected to have experience in research. So don't have to "cover for research experience".

> (iii) Rethink the whole PhD thing and get a professional degree like an MBA (did well on GMAT)

And, with that MBA, what will you do with it? Consulting? Try to work your way up as a C-level guy? Try to do portfolio management or be a venture capitalist?

I'm a former MBA prof -- typically an MBA is not very technical, but, now, especially with some much in computing, being technical can be seen as an advantage. Or, maybe the flip side is, soon all the work that the techies can do will be done and what will be left is the non-technical work. Your guess.

As far as I can see, still need to hustle and/or be lucky to have a good career.

Also, an MBA is expensive while a Ph.D. often costs $0.00 for tuition and might provide a stipend.

> (iv) Just continue to learn on the side and try to use my skills for opportunities

That can be a short-term approach that can lead to a degree later.

If you sense that your background is not good enough to go for a degree, then, sure.

A major cause of my success in grad school is the math I studied independently between my Bachelor's and my Ph.D. program.

In particular, for a Ph.D., will likely have to pass qualifying exams, and you will want to have enough preparation for a path to do that.

> How did you narrow your interests before applying for a PhD?

I attacked some practical problems with applied math and computing. I did this, first, in work in US national security around DC.

> Is it possible to get a masters level pure math education through self study? I've taken until PhD level measure theory.

IIRC at one time the Web site of the math department at Princeton said that the graduate courses were introductions to research by experts in their fields, that no courses were offered for preparation for the qualifying exams, and that students were expected to prepare via independent study. For this, you will need a good undergraduate background.

There is some question how much a student or anyone should attempt via independent study. As a researcher, a lot of independent study is usually just part of the work, but there are seminars, etc. that can help.

Mostly I did well with independent study, but once in statistics I picked a poor book and wasted some time. And, for some subjects, I did get and really needed the good, mature overview of a good course -- else there are too many places along the way where can waste time, get stuck, or go off on a tangent. Also it can be good to be around good researchers and learn their attitudes and approaches: E.g., often researchers go through new material surprisingly quickly and, really, without the thorough mastery often implied as important or even essential in high school and undergraduate school. E.g., in research, for some new material are trying to make use of it, not get a high score on some GRE exams.

Thanks!

A lot can change over the last 35 years : http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5517&from=fu...

There will be some issues of social maturity that can get in the way of the research. E.g., typically an undergrad student is still under some influence of some of the unfortunate attitudes of some of their high school teachers.

E.g., after my junior year, I had a summer job in a bio-physics lab doing electronics. I knew Maxwell's equations well enough, but what I didn't know, and hurt a lot, was that I really needed to get the data sheets on some power transistors, read off some values, and, then, just apply Ohm's Law to characterize the usual operating range of those power transistors. I didn't have enough social maturity to do well enough asking others, and from high school I was too afraid just to use my own judgment, strike out into the unknown, and discover that all I had to do was just use the data sheets. That is, I guessed that digging into the data sheets would be too much of a detour and I'd get criticized. Also the guy I was working for kept emphasizing that I should take a practical approach and shouldn't go off into too much theory. Gee, I just needed to read the darned data sheets and, there, the curves that showed the basic, simple minded performance of the power transistors.

Out of undergraduate school and in a job for a while and with some good successes, I came to have a sense of what where good, appropriate, and promising investments into the unknown. The lessons of high school conflicted with that "sense".

Why would you assume I would bring up wet lab work in the context of mathematics? https://sites.google.com/site/mathreuprograms/

Here's a 20-year-old 20-year retrospective: http://www.d.umn.edu/~jgallian/tanglewood.html

I was asking.

Okay. Graph theory, etc.

I'm not writing a thesis, but a book on teaching coding, and this one point is really important. One of my chapters was on the cognitive benefits of writing software code--which I assumed were great because of Seymor Papert ("Mindstorms") and Ted Nelson's ("Computer Lib/Dream Machines") glowing enthusiasm for teaching kids how to code.

I started writing the chapter and had to get into the literature to find references to support the idea that learning to code carried concrete cognitive benefits. The research was extremely mixed, with one survey of the literature being highly critical of Papert for assuming so many benefits when the research was not finding that at all. I ended up having to stop everything and just read papers for a week to understand what science really knew about the subject. When I went back to rewrite my chapter, I had to temper my own enthusiasm and add numerous qualifications and cautions about my claims considering the evidence.

I'm happy for the experience as my thoughts on the subject are much more highly nuanced now, but it was very disheartening at first.

I tend to try to solve a problem myself first (this could be days or weeks), then look at the literature. If I looked at the literature first, then my ideas would look like the literature.

It's a gamble either way

I maintain that the most important domain knowledge is understanding what evidence/methods are actually robust, and what just sounds nice.

But yes, any research requires a lot of reading. If you don't like doing that (and ideally you really enjoy it) you're going to have a very tough time of it. You also won't have the background to have good conversations with people or to design experiments.

I wish I had known to ask that sort of thing. My advisor typically kept a lab of a dozen postdocs and a single PhD at any given time. I think in 25 years of being a research professor with sizeable grants that advisor only graduated 4 doctoral students, and a rather distressing proportion of the postdocs left not only academia but science after that lab.

I would also try to hang out with the current doctoral students and assess their psychological well-being.

I was able to successfully complete my PhD, but I took more time than expected. Moreover, I was no way near as productive as I'd hoped to.

Some people can actually do very well in what appear to be hostile labs. Some needs lots of space, while others do best with close monitoring. It ultimately just comes down to whether the advising relationship works or not, and it's nearly as complicated as romantic relationships.

One thing that I like to stress is that you don't have to like your advisor. You should probably not dislike them, but what matters is how you play off each other and work together.

An example: some advisors will put the people in their lab in somewhat adversarial positions. There will be two or more people working on similar projects, and the more successful of them will start to get more and more of it. For certain types, this situation is actually very motivating and can produce two differentiated projects that support each other. For others, it will undermine their confidence and wellbeing to a terrible degree. There can be a lot of resentment among both parties.

While I would generally consider this a warning sign, you must recognize that this can be a good environment for the right person. The difficulty is that it's hard for you to know what works for you until you do it. There's no best formula, you just have to choose as best you can.

That is, in principle all points make sense, but they are either truism, too obvious, too vague or things that are out of control. (If I had read it before starting my PhD it wouldn't have changed a thing).

Or maybe I am overly skeptical of simplistic life-advice? Is there any take-home message that changed your way of acting?

You should fix a lot of other things first (in particular, prefered methodology and advisor) and then, after fixing those things, choose applications toward the left-hand side. A few reasons:

1. If you're doing it for hype alone you will be miserable. Ph.D. students tend to be out ahead of pretty much everyone in terms of the nitty gritty details, so they often hit the trough of disillusionment right around the time everyone else is at the top of the hype curve.

2. A Ph.D. is a once-in-a-lifetime opportunity to be paid (sometimes comfortably) to work on (mostly) whatever you want. If it's not at least a little self-indulgent, you're wasting a great opportunity.

3. You'll have the rest of your life to chase the next sexy thing.

A PhD demonstrates that you can successfully conduct original research. Period.

The time for creativity and really making a name for yourself is in a post-doc. That's when you differentiate yourself. You can do this during doctoral work, but it is far from required. As long as you're doing something significant (which any lab you join should be doing), you'll likely encounter plenty of new insights that really do contribute in a novel way.

This can let you focus on process and methods that lead to good results. You learn to troubleshoot, to adequately record observations, and to be able to write all that up in a coherent and thorough way. These are the key lessons, and those most valuable and translatable outside academia should you take that route.

In the end, I think that creativity comes easily and naturally if you're on top of the literature and actually performing experiments. If you trust your observations (which takes a lot of effort to get there) and are looking at interesting things, you will set down interesting paths of inquiry.

I concur wholeheartedly, and this is my recommendation to every new PhD student. Replicating earlier work has many benefits for a PhD student in the beginning of their studies:

* It forces them to deeply understand the system they're replicating. Much more deeply than simply reading the paper would.

* It allows them to uncover the many corner cases which the original paper probably just briefly mentions. (If it mentions them at all.)

* It makes them think about how they would design an experiment so that it is repeatable and reproducible.

* It is a good source for inspiration for alternative solutions to the problem or even new problems altogether.

Alas, replication is way too uncommon in systems CS, which I think is one of the reasons why other scientists often look down on systems research.

I would say that this danger applies more to those doing practical (applied) work. A practically oriented program will often have a clear goal, and preset goals are often unattainable in research. By contrast, a student doing theoretical work can always bend their goals when necessary.

http://henrikwarne.com/2016/03/07/ph-d-or-professional-progr...

Finishing a PhD is unlike completing a project in most other jobs. In most other jobs, someone needs what you're working on. Other people's investment in the outcome is similar to your own. If you fail to complete your work, others are likely to fail to complete their work. Consequently, incentives (hopefully) line up, and infrastructure (hopefully) exists to support you, with the recognition that your success is linked to group success.

Your PhD dissertation is not like this. Yes, your adviser is invested in you finishing - but not as nearly as much as you are. They will have other students, and they can always work on their own. Your peers may be invested, if they are working on another part of the project - but if you do not finish, they will find a way to get on without you. Your university is invested in you (quite literally, most of the time, with money), but again, not as nearly as invested as you are: plenty of grad students never finish, and they will help you, but schools also recognize that not all students finish.

The author has a good list, and I may read his book, but he's missing this attitude that I felt I had to adopt. The person who cares most about you finishing is you, and sometimes that means having to be selfish in order to finish. That can take of the form of not engaging in as much service in your department, or not providing some help on a project that is not part of your dissertation.

I do think this attitude is unfortunate, but it is a natural result of the requirement that a PhD dissertation represents work that the student owns. I much more enjoy the research I have done in an industry research lab, where me and my colleagues have collaborated equally. (Or equal-enough that in a grad school context, no one person could claim ownership of the work for a dissertation.) But, it's the system we have, and because of that, I think that in order to finish, grad students have to - at least eventually - adopt a selfish attitude.

Specifically, this "selfish attitude" means ruthless evaluating: will this thing get me closer to graduation? If no, don't do it. (Obviously this only applies to work. Having a life outside of grad school work is enormously important.) In the beginning, I don't think one needs to have this attitude. But as you approach completion, I think one needs to start thinking this way.

The people who drop out of PhD programs usually are plenty smart enough, they just don't know how to make things happen.

Then again, some people seem content and happy to do 8 years of PhD studies followed by 10 more of meandering post-docs. Bummer is when they're surprised that no one wants to make them a professor.

Universities also exploit people who depend on academic success for self-worth and validation. The head games that advisors play to goad graduate students on to do more work can be pretty unscrupulous. The very fact that universities can convince some of the smartest (if not those most possessing in common sense) people to work long hours for below minimum wage, while learning a body of knowledge of dubious value, is pretty amazing when I look back at it.

In the real world, people pay you good money to do PhD level work. Job security and low work loads aren't on offer if you do an expected value calculation factoring in all of the tenure washouts. The only thing graduate programs offer in return for collapsing highly educated wages is an infinitesimal chance at fame. It is no less an exploitative championship system than professional sports or Hollywood.

I think the only good reason to seek a PhD is for altruistic reasons, to expand human knowledge. But keep in mind that you are choosing the life of a 21st century cleric, with its attendant material deprivation and very good chance of never marrying or having a family.

    The people who drop out of PhD programs usually are plenty smart enough, they just don't know how to make things happen.

Generalizations like this aren't useful. I could just as easily say "The people who finish PhD programs are plenty smart, they just don't understand the sunk costs fallacy."

I dropped out to make things happen. I took a job where I learned and developed myself more than I did as a student.

It's possible that those of us who drop out of PhD programs have, in fact, made quite a few things "happen" in our lives and realized that the job market was terrible, the subject was... just stupid... and that there are other things in life than petty academic squabbles.

Just to disclose, I was about 6 months away from defending my dissertation and didn't want to throw another half a year of my life away working at a minimum wage job so I might (possibly) be able to work at a poorly paying job for the rest of my life.

It was the first really great adult decision that I made in my life.

I don't think that's fair either.