Part I. I'll try to give an answer and a response more generally to the material in the OP. Background. I got a BS in pure math with nearly a second major in

Part I. I'll try to give an answer and a response more generally to the material in the OP. Background. I got a BS in pure math with nearly a second major in physics and worked in computing and applied math for problems in US national security around DC. Jobs were very easy to get; at one time my annual salary was 6 times what a new high end Camaro cost; much of the work was challenging for both the computing and the applied math; I was learning a lot of both computing, e.g., algorithms in Knuth, and applied math on the job and also especially math in independent study on evenings and weekends. Soon I got a call from a college friend to join FedEx. I used my background in computing, with a little applied math, to write some software, in six weeks, to schedule the fleet. The results pleased the BoD, enabled crucial funding, and saved the company. Later the BoD wanted some revenue projections. I formulated and solved y'(t) = k y(t) (b - y(t)) for time t, revenue y(t) at time t, rate of growth y'(t) = d/dt y(t) at time t, and full revenue potential b. The BoD was pleased, and ..., to make a long story short, I saved the company again. Then I got a Ph.D. in applied math from the engineering school of a famous, high end research university. Now I'm doing an Internet startup, a Web site, basically a new and very different search engine -- for the, IMHO, very large part of search handled at best poorly by the existing Web sites and well known techniques. For the startup, my applied math background is crucial: The crucial, enabling core of the startup is some original applied math I derived based on some advanced pure math prerequisites I got both in grad school and in independent study. I already knew enough computing except had to learn how to bring up a Web site that has been easy for the user interface but due to the core math somewhat tricky on the server side. I learned Microsoft's Visual Basic .NET (for the programming language -- I like it), ADO.NET (for the Web pages), and ASP.NET for the (relatively meager) use of SQL Server. The main difficulty was working through 5000+ Web pages of documentation. The first code is the first production code, 100,000 lines of typing, 24,000 programming language statements, and lots of documentation. The code seems to run as intended, but I need to add some data. Some lessons: (1) Math. IMHO, the key to powerful, valuable, new applications of computing is applied math. That is, if we accept that the big opportunity is to exploit and apply current computing, then we might notice that whatever we code to put out data users will like, as information, entertainment, whatever, is necessarily mathematically something, understood or not, powerful and valuable or not. So, in some sense, from 100,000 feet up, it should help to proceed mathematically, with possibly advanced prerequisites, some new results focused on the application in mind, and complete with theorems and proofs. Just IMHO. But I don't know anyone with a yacht over 100' long that did that; I suspect that very few people agree with me. Maybe what I am saying is a hopeless wild goose chase or a great green field opportunity -- you judge. (2) Getting a Ph.D. In a nutshell, the three most important parts of getting a Ph.D., in no particular order, are research, research, and research. Yes, there can be courses, credits, grades, teaching assistant positions, weekly research seminars, qualifying exams, etc., but at a research university what can "cut through", dominate, and trump all or nearly all of that is research. The research should be publishable in a good peer-reviewed journal of original research; if there is any question, then send it in. The criterion for a Ph.D. dissertation may be something like "An original contribution to knowledge worthy of publication." -- so, in case of some doubt, publish the thing. The usual criteria for publication are that the work be (i) new, (ii) correct, and (iii) significant. Now for a non-standard observation and recommendation: Go for applied math in an engineering school. Start with a real problem, hopefully a significant real problem, likely from outside academics, hopefully identified before or early in the Ph.D. program. Do some new math to get the first good or a much better solution to the real problem. So, the math is "new" -- got (i)! Since the work is math, with theorems and proofs, it's easy enough to check for "correct" -- got (ii)! Since have the first good or much better solution for the real problem which is hopefully significant, get "significant" -- got (iii). Note: The math may not, just as stand alone pure math, be seen as significant -- so, likely have not proved the Riemann hypothesis, shown that P = NP, etc. But compared with a lot of pure math, are already ahead by one point -- have one real world application! In my case, I started with a problem I had identified in industry before grad school. I found an intuitive and rough solution on an airplane ride. In my first year in grad school, one course I took let me make solid math out of my intuitive solution; I did that independently in my first summer, walked out of the library with an 80 page manuscript that had all the actual research for my dissertation. Then I encountered some of the nasty nonsense as in the OP: There was a prof who didn't like me. He thought of rows, columns, and layers, lines, stay within the lines, rules, etc. and resented that I'd basically written my dissertation independently within 12 months of arriving on campus and before taking the qualifying exams. Well, a course had a question but no answer. I did a good enough literature search and saw that likely there was no answer known -- since it was a very narrow question, it was easy to do the literature search. I got a reading course approved to address the question and write a paper, maybe just expository and maybe without a solution. Just before getting the reading course approved, in a few evenings I found a rough solution. So, got the course approved -- shook hands with the prof. Then I cleaned up my first solution, mostly sitting beside my wife on our bed while she watched TV and I worked on the problem, and found a much better solution and a general result that was surprising, even shocking, and settled some related questions. That took two weeks into the reading course. I submitted my manuscript of about 20 pages, and I was done with the reading course. Fast course. The course also had three credits and gave me the last credits I needed for an MS. News of my work spread through the department; my favorite prof walked up to me in the hall, "I heard about your result. It also says that ....". Yup, clearly it did. The result was clearly publishable; later I did publish it -- no problem, accepted right away. The biggie, practical result was that suddenly I had a halo and a coat of Kevlar armor against any criticism; any of the faculty would have loved to have done what I did. To defend against the abuse as in the OP, I recommend -- do some publishable RESEARCH. (3) Handling Ph.D. Qualifying Exams. At least at one time, the Web page of the math department at Princeton stated, IIRC (if I remember correctly): "Students are expected to prepare for the qualifying exams on their own. Graduate courses are introductions to research by experts in their fields. No courses are given for preparation for the qualifying exams." In part, the qualifying exams are like a foot race, but in this race you can get a head start and be 1 foot from the finish line when the starting gun goes off. I consider that Princeton policy to be somewhat wise. So, prepare for the qualifying exams on your own (to be able to do this in math, a good ugrad pure math major should be sufficient to let you know how to study and learn, make good progress, and not get stuck or lost). To do this preparation, get the best, focused information you can on what will be asked. So, get recommended texts, copies of old exams, maybe chat with some profs and some students who have taken, hopefully passed, the exams, syllabi of any relevant courses, etc., maybe all before applying to the Ph.D. program. Then study. Use some judgment on how deep to cut and how many proofs to memorize -- cut deep enough but not so deep you take too long or just quit. Then show up as a first year Ph.D. student, do well in some courses, do well on the qualifying exams, complete your research, listen to Pomp and Circumstance, get your degree, and LEAVE. (4) Academic Career. If you want an academic career, then maybe don't leave the grad program so soon. Instead, publish some papers, get some streams of promising research going, meet people at research seminars and conferences, get known by the Editors in Chief of the journals or conferences where you publish, if invited to give a talk at a conference, do so, do the usual meet and greet and publicity, build your own professional network, etc., hopefully some of your profs will help you get some job interviews, etc. Learn how the academic games are played -- there are some really important academic games, and you very much should learn how they are played. Then you will be in an academic career. One possible prof slot is in a B-school. Consider that. Someone with a good applied math background has a heck of an advantage in a B-school. Papers that make progress on practical problems are commonly considered good research in B-schools. People who want to hire consultants tend to regard B-school profs as more practical , i.e., motivated by money, than more pure profs.