Changing Careers: Becoming a Professional Mathematician First, I apologize if this question is too soft or doesn't comport precisely with what is considered a good question, but I know of no other place to ask it.
A little background: I obtained an undergraduate degree in mathematics about 20 years ago. I was near the top of my class, however at an admittedly sub-par school. I have, on-and-off, studied mathematics independently in my spare time in the intervening years and although I have undoubtedly forgotten some technical details (such as specific integration techniques, various methods of solving differential equations, etc.), in general I have expanded my knowledge considerably beyond what I learned as an undergraduate. I have particular interests in differential geometry and algebra and feel I could be successful studying these topics and other mathematics seriously at the graduate level. 
I am a professional software engineer specializing in business application development and, honestly, I'm bored to tears with my work. I have pondered over the years going back to school and working toward a PhD in mathematics and I have recently been giving this more thought. Based on my understanding of the situation, upon successful completion of such an undertaking, I understand that there would be little hope for any research (or teaching, for that matter) positions in academia. However, there seems to be a strong demand for virtually any STEM-literate people in industry so I would expect to be able to find work as a mathematician in some capacity somewhere.
So, one question is whether this latter expectation is realistic. Specifically, if one obtains a PhD in pure mathematics is obtaining a mathematically-oriented position somewhere in industry afterwards a realistic goal? Yes, I realize that Google, MS and other software companies hire Math PhD's, but I really want to get away from this particular field if possible.
My next, and more important, question is if I decide to go down this path how I could ever get accepted to a reasonably-good school? I understand that I would need to do well on the GRE subject test and I believe I could manage that. The fact that I did well as an undergraduate would probably also help but the fact that my degree isn't from an especially well-regarded school wouldn't help. One of the main stumbling blocks would be the (usually three) required letters of recommendation. Now, it would be all but impossible for me to provide these - as I said, its been 2 decades since I was in school. So, given good performance as an undergraduate and a reasonably good score on the GRE, is there a way to get around not having recommendation letters? I suppose I could get such letters from recent employers but I doubt this is what graduate committees are looking for.
Thanks for taking the time to read this. 
 A: I'm an assistant professor at the University of South Carolina. Although we are not Harvard or Princeton, we are a good research department with a solid Ph.D. program, and many of our graduates are successful both within and outside academia. Several of our graduate students are older than I am, which I think is great!
The graduate admissions committee will look for firm evidence that you are prepared to succeed in graduate school. In particular, you will need evidence that you currently have mathematical ability and motivation comparable to other entering grad students. To this end, I don't really see any way around getting recommendation letters. 
I strongly second Suvrit's advice -- I was out of school for a couple of years before grad school, and I did the same thing. I would also encourage you to do the homework, take the exams (if any), and ask the professor to take at least a brief look at your work. Although the professor is not obligated to grade it, I think many would be willing to do this. If you do a good job, your professor will be able to write you a good letter (and also give you useful advice).
I might add that you will want to demonstrate knowledge of foundational topics, in particular algebra and analysis. If the courses you sit in on aren't in these topics, or  don't have them as prerequisites, you might ask your letter writers to spend a half hour vetting your background knowledge so they can address it in their letters. 
Good luck!
A: Have a look at: Too old for advanced mathematics?
There is a lot of good related advice there, that might help you think through your case a bit. 
Also, before you jump into the new thing, do the following simple thing:


*

*Goto a nearby university

*Sit in some math classes, both undergrad and grad

*Make connections with professors there, and then seek their advice and mentorship on how to proceed -- once they know you a bit, they will probably be in a much better position to advice you than a website like MO.


Good luck! One is never too late to enjoy the pleasures of math.
A: You asked two questions. First, jobs you could hope to get. You know that the market for recent PhDs in academia is bleak, but that doesn't mean there are no jobs. If you get accepted to a reasonable graduate program and find yourself a good advisor (this is more important), then there should be some hope of jobs in academia at the end of the tunnel. They might be jobs with high teaching loads, or jobs in locations you don't want to live, but they do exist. Have you thought about how much you're willing to pick up everything and move? Have you thought about whether or not you'd be okay teaching introductory calculus over and over again to students who don't care all that much?
Let's talk about jobs outside academia. The biggest employer of mathematicians in the world is the National Security Agency. They employ research mathematicians and would view any knowledge of computer science as a massive bonus. I believe that if you made it clear to them that you wanted to do mathematics research then they wouldn't force you to do software engineering. You still wouldn't get to pick your own problems, but by now you are used to that. I'm sure the NSA has a wide variety of problems you could work on, and that you could find something sufficiently mathematical, but where a good understanding of theoretical computer science, computer hardware, etc would be an asset. Cryptography is a nice way to combine algebra and computer science, and if you could get a job in Crypt then you could probably do as much algebra as you want. There are also surprising ways to use Differential Geometry in real-world applications, after representing a data set of interest as a bunch of vectors in some giant space. There are also a ton of NSA contractors, the biggest of which is the IDA (Institute for Defense Analyses). If you do some googling, you can find a large pool of employers here, and many will employ you even if you just have a masters degree in mathematics.
Another place which no one has discussed in depth yet is Microsoft Research. The Boston office in particular is very good and has employed many amazing pure mathematicians (e.g. Laszlo Lovasz, Vera Sos, etc). They tend to be graph theorists, but they have complete freedom to work on whatever they want. There are other employees whose sole job it is to figure out ways to apply what the mathematicians are doing. The person running this branch is a very good mathematician named Jennifer Chayes, and I learned everything I wrote in this paragraph from a talk I saw her give. The problem here is that they seem to only hire rock-stars for full-time positions. It's a bit easier to get a summer internship there (even for non-graph-theorists; I know an algebraic geometer who got one), and you might be able to parlay that into a job. But you'd need to be in grad school at the time.
Hedge funds are starting to hire mathematicians again. This is also a profession which would like someone with a background in software engineering. I believe one thing mathematicians do at Hedge funds is find ways to do simple computer operations faster. Unfortunately, this would probably be a place where you wouldn't be using the algebra or differential geometry you studied in grad school. Have you ever considered a career in finance?
Next, how to get into graduate school. I'm a graduate student at Wesleyan University, which has a tradition of taking on non-traditional students. When I got here we had a 65 year old man as a PhD student. You want to find a place like that. I think Wesleyan is moving in a different direction now, but I believe Montana State University is another which might work. To be such a student, you need to have great letters of recommendation, but they probably don't have to be from famous research mathematicians. I agree with Suvrit that you should take courses at a University near you, and try to get both advice about which programs to apply to and a good letter of recommendation out of that. If your other letters are from people you've worked for, and they talk about your work-ethic, your attention to detail, your mental toughness, your drive, etc then I think you'll be fine.
Many undergraduates have sought my advice about graduate school over the years. I always try to get at their real motivations. Being bored with your current job is not enough to get you through grad school. The only really acceptable reason to go is if you love learning mathematics, problem solving, and doing research (or if you have a strong reason to think you will love it when you get to it). Going to grad school just so you have a credential to open up new job avenues is a sure-fire way to either drop out or end up in a job where you are a slave to your teaching. It might be worth considering getting a masters somewhere first and then deciding if you really want the PhD. It's a big time commitment and during that time you're making almost no money. At your age, I feel like this would be a strong disincentive. With a masters degree a lot of doors are opened up (e.g. NSA) which might not be open right now because you've been out of math for so long. Here are 2 good articles about grad school, which should help you sort out your real motivations and see if it's the right thing: first, second. Two good books for trying to get you through it are: Krantz, Peters (not all the advice in here is good, some is for non-mathematicians only).
To end on a positive note, if your soul-searching tells you that grad school is right for you and you can't truly be happy doing anything other than mathematical research, then clearly you have to go. If that comes about, let me just say that I think that your years outside of academia will give you a very valuable perspective and motivation. I've noticed that grad students who worked for a few years after undergrad usually came back for good reasons and with great work-habits. They were able to use those to get through the down times while those who came directly from undergrad struggled because they were never sure if the grass on the other side was greener. I'd encourage you to really own those years you spent outside academia and not see them as a mistake when you're working alongside much younger people. Plus, if you do apply for jobs at small liberal arts schools after graduating, they might like you more for the perspective you bring and your increased ability to advise undergrads.
Good luck!
A: Alan, I have been in a situation very similar to yours.  After getting my Bachelor's in math (also from a small, essentially no-name school), I spent 23 years in the software industry (Silicon Valley).  Boredom and frustration with my work, and an ongoing interest in math since my undergraduate days all motivated me to consider returning graduate school.  Being laid off during the recession in 2002 gave me the final nudge.  I immediately entered a master's degree program in math at Cal State Hayward, full time.
At first, I was thinking that the master's would be my terminal degree.  It was only after I gained a mentor while in the program that I learned that it was feasible to go on to a PhD program, even at my age. In particular, the vast majority of PhD grad students in math have a teaching assistantship, which pays for tuition and supplies a (very modest) stipend.  One bit of advice to me was to not go to any school which did not offer a TA-ship.  Working on my master's provided me recent reference letters. (Some of my best undergraduate references were deceased.)  I cannot say that working on a master's before the PhD program is required, or even recommended.  Be aware that many schools have programs where a student enters under a master's degree program, and then before completing the master's the student can decide, together with faculty consultation, whether to continue on to a PhD program.  (My PhD school, the University of Washington, offers such a program.)  In fact a few schools require all their students to complete a master's before formally being accepted into their PhD program.  These programs, where master's and PhD are both done at the same school, will save some time overall since all units are guaranteed to be transferable. (None of my master's degree work at CSUH transfered to UW, but it did equip me to pass one of the preliminary exams at UW.)
I finished my PhD in 2009.  I was hoping to enter academia, and I did a 2-year postdoc at UBC.  But as everyone has noted, the academic market is dismal at this time, and I took a position in government.  So it is good that you are aware of the job market conditions and planning accordingly.  As David White noted, there are several industries that hire mathematicians.  He also noted that the NSA and affiliated contractors hire a great number of mathematicians, doing interesting work. I would also note that there are other government labs and agencies that also hire some number of mathematicians.  In any of these, I would suggest that while it's OK to seek "math-oriented" work, you should at the same time remain open to leveraging your industry experience, especially any software skills you may have acquired. 
All in all, I have no regrets for pursuing a PhD.  Indeed, I would have had regrets had I not.  Best to you in your pursuits.
A: You may want to do a little more self-searching: why do you want to become a professional mathematician?  Do you really want to be paid for doing mathematics?  Or is it that you want to do mathematics and also want to pay the bills?
There are many costs to doing mathematics professionally.  In my case I saw the potential for a high cost involving interoffice or departmental politics that I wanted to avoid, so I chose not to pursue an academic career.  The other costs you can imagine or inquire about separately.  You can probably infer what the costs are for doing mathematics in an industrial situation, such as the pressure to produce results at a business pace.
If you decide that you do want to find an industrial niche in which to do mathematics, I recommend talking to the employees and employers to find out what it takes to get there.  Some employers may have a great enough need that they will fund your way through school.
I have other suggestions, but they should wait until you have a specific and well-thought-out goal.  (You might have that already, but I do not glean that from your post.)
Gerhard. "Ask Me About Amateur Mathematics" Paseman, 2012.02.11 
A: In my day a high score on the GRE subject test was necessary but not sufficient for admission to graduate school.   Letters of recommendation are also very important.  These do not need to be from researchers in mathematics; letters from employers should be fine.  However, the letters do need to discuss your suitability for graduate school; letters that discuss your mathematical ability are particularly useful. 
A: Currently I'm a PhD student in applied math at a fairly good research university. I just wish to share some of my experiences which might help you make the decision.
My undergrad education was in Electronics Engineering. That's when I truly fell in love with math. I did extremely well on math courses and extremely poorly on the rest, decided that I was best suited for math and decided to pursue grad studies in math. After my bachelor's degree, I spent about a year and a half studying undergrad math (mostly advanced calculus, analysis, ODE's, linear algebra and modern algebra). I took the GRE subject test and did average on that. Looking back, I realize now that I should have spent at least a year more making more intense preparations. But back then, I was working alone all by myself, without anybody to guide me through my difficulties, and without any clear idea about the overall picture and structure of mathematics.
I was able to get admission into MS applied math program in my current university. At the end of my 1st year as an MS student, I had passed both my PhD qualifiers and decided to go for PhD. That has turned out to be the worst decision I have ever made in my life.
The problem was that I was, and I still am, in love with pure math and I overlooked the fundamental differences between my own intellectual preferences and the research work that was going on in our department. Although our applied math dept is quite renowned, it does not contain any professor who has a PhD in mathematics. Most of them are physicists, some of them are biologists, some computer scientists and some statisticians. Now, I have all the respect in the world for these disciplines, but in the end my work turned too far away from real mathematics. Today I spend about 95% of my research time doing C/C++ coding, about 3-4% physics and the rest is math.
There was a time when I was seriously under the impression that I was going to become a professional mathematician at the end of my PhD. However, a few years down the road, I realized that I am not even close to being a mathematician. It became clear that if I want to become a professional mathematician, I'm going to have to pursue another PhD, this time in pure math without any compromises.
Today I'm carrying on with my PhD in applied math only because I have spent too much of time and effort to turn away from it. And I have full intention of going for a second PhD in pure math. I have learned some of the biggest pitfalls in this process, and I have learned them the hardest way possible.
During all this ordeal, my love for math, especially theorem-proof type math and problem solving in general, has grown tenfolds. And I realize today that I will never be happy without mathematics. I am almost 31 today, but I'm determined to go for pure math.
My experience says that you don't necessarily become slow as you grow old, but it certainly becomes difficult to commit yourself 100% to math. Especially if you have a family (as I do), and if your family members don't understand your passion for math, then things can get a little rocky at times. But that next proof, that next eureka moment, that next beautiful problem make it all worth it!
