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Part I I will give an answer focused on STEM field Ph.D. programs in the US. Pros: Professor Career. A Ph.D. is nearly essential for a career as a professor

• graycat 13y ago
  Part II (6) Background. A good way to win a 100 yard dash foot race is to start one foot from the finish line. Otherwise, get a head start. Or, get an 'unfair advantage'. So, for the approach of 'mathematizing' engineering, start with a solid undergraduate degree in pure and applied math. Then get a Master's concentrating on selected topics in pure and applied math. E.g., even if you want to get a Ph.D. in computer science, as an undergraduate, mostly just f'get about the undergraduate computer science courses and be a math major instead. The undergraduate computer science material won't much help you in computer science research, but the math can be an overwhelmingly strong advantage. As a math major, after the usual undergraduate calculus sequence, take theorem proving courses in abstract algebra and linear algebra -- the first is useful at times, maybe more so in the future, and the easy place to learn to prove theorems, and the second is likely the most important material in mathematical analysis and applied math. To continue, take a second course in linear algebra. A good start is the now classic P. Halmos, 'Finite Dimensional Vector Spaces'. It was written in 1942 when Halmos, a fresh Ph.D. from Doob at U. IL, was an assistant to von Neumann at the Institute in Princeton and is really a finite dimensional introduction to von Neumann's Hilbert space. For more, see the books by R. Bellman or R. Horn. Work carefully through W. Rudin, 'Principles of Mathematical Analysis.' Supplement that with Fleming, 'Functions of Several Variables', Buck, 'Advanced Calculus', etc. As a crucial start on optimization, work carefully through, say, Chvatal, 'Linear Programming'. Somewhere get a good theorem proving course in ordinary differential equations; that can be a good start in a lot in applied math and on deterministic optimal control theory. At one of the first chances, take a careful pass through Royden, 'Real Analysis' and the real half of Rudin, 'Real and Complex Analysis' -- a good course would help weight the more and less important ideas there. Then a long dessert buffet is Luenberger, 'Optimization by Vector Space Techniques'. With that background, you can do probability and stochastic processes the serious way via the more serious authors, e.g., L. Breiman, D. Brillinger, K. Chung, E. Cinlar, J. Doob, E. Dynkin, I. Karatzas, M. Loeve, J. Neveu, S. Shreve, and more. For more, attend research seminars: Don't often try to follow much of the content but just use the seminars as suggestions for new fields, problems, techniques, names, references, etc. Also you may meet some people or learn about career opportunities that could be helpful. If after a Bachelor's or Master's want to drop out of school for a while, then do so, get a job, stay single, and outside of work lead a simple life and study some more pure and applied math from some of the best sources. Hopefully get some practice applying math to real problems. Get a collection of real problems that might use for your Ph.D. research (remember, your Ph.D. is to be in engineering). Make some first cut progress on those problems. If you can get some solutions, then write them up as papers. If some of your papers look publishable, then try to publish them. (7) Ph.D. Program. Now apply to a Ph.D. program. Include your papers, especially the published ones. Your mathematical background can be a huge advantage. In particular, consider working in optimization and stochastic processes with a solid background in measure theory and functional analysis. Heck, nearly everything going on in a big application, a server farm, a large network, ..., the economy is a stochastic process where we want to optimize. Do what the department insists on, and otherwise continue your research. Take your best research and submit it as your Ph.D. dissertation. If there is any question about the quality of that research, then publish it; or just submit work that you have already published. That the work was accepted for publication in a good journal tends quickly to settle all doubts about sufficient quality. In particular, proceeding in your Ph.D. program as outlined here, you have (A) obtained your own background in pure and applied math, not depended on your department for that background, and have quite likely obtained a better background than any of your professors; (B) have selected your own research problems and not depended on your department or its professors for research problems; (C) have done the core of your research in applied math with theorems and proofs which are comparatively easy to show are correct and difficult to criticize; and (D) have used publication of your work preemptively to establish a respected, outside, objective proof of quality. (8) How to Do the Research. Mostly the mathematics is just math and not very intuitive and is solid and not just guessing. But finding that math, that is, original math, can use quite a lot of intuition and guessing for finding what might be true and finding ways to prove it is true. My view is that the most important work in math research is intuitive with a lot of guessing, a lot of simple models, and a lot of simple, intuitive, testing of intuitive guesses. (9) University. Generally you are better off at the best research university you can get into. The less good universities can force you to jump through no end of silly hoops and be so insecure in their own expertise as to delay and delay saying that your work is good. (10) Courses. At a really good university, it may be that the graduate courses are not much like undergraduate courses and, instead, are essentially just introductions to narrow parts of research by experts in those parts and, really, intended only for students wishing to pursue research in those parts. (11) Done. Accept your Ph.D. and go do something else. Your professors may have helped you get a job, if so likely an academic job; thank them, and if that job is what you want, then take it; else proceed with your career along lines you've had in mind. (12) Warning. A Ph.D. program can be dangerous, harmful to you and your life and career and even fatal. If you are not well protected with your own background in, say, math, your own research problems, your own research, and your own publications, and hopefully your own financial means, then your education and much of your life can be in the hands of others who can be clumsy, competitive, nasty, arrogant, domineering, abusive, destructive, sadistic, incompetent, etc. It can be that, really, your professors don't have any good research problems for you. E.g., even in an engineering school, they may have nearly no contact with real problems from off campus and, thus, little or no help in finding a real problem for your start. Your background in each of real problems, math, computing, and business can easily be much better than theirs. Each of your professors may have been beating their head against some hard problem for the last 15 years while you have some good insight into some good, new problems you have a good chance of solving. It is accepted that one of the keys to success in research is good problem selection. You can feel that you are in jail without being accused of a crime, with an indeterminate sentence, tortured by your professors as jailers, and with no chance of parole. There is a special warning for students who made Valedictorian in high school and PBK, 'Summa Cum Laude', etc. in college. In the text version of 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." So, a student who has done really well based mostly on fantastic memory, pleasing the teachers, dotting i's and crossing t's, doing just what was requested, working desperately for praise and approval from others, terrified of any chance of criticism, out to 'change the world' in major ways or bust, can find themselves in a situation of inhuman stress, then depression, then incapacitation, then more stress, then clinical depression, then death. No joke.