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.

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