Applying to a PhD in Applied Mathematics - Looking for Advice

[Axion004]

I am looking to start a PhD program in applied mathematics during the Fall of 2018. I have some realistic concerns that I would like to vent on this forum. I created this post because I am unfamiliar with academia and don't want to create a plan that would lead to failure (my personality is INTJ).

Background: I am a normal student who is quiet and studious. After I obtained my undergraduate degree, I worked for five years as a software consultant. During those five years, I have a gained a lot of experience in mathematics by reading books and studying math in my spare time. My undergraduate degree is in an unrelated subject, I have a MS degree in applied mathematics. I have taken a fair amount of courses relevant to my goal. These include calculus 1-3, introductory physics 1-3, linear algebra, advanced linear algebra, ordinary differential equations, partial differential equations (two courses), mathematical modeling, complex analysis, thermodynamics, object orientated programming, data structures, econometrics, dynamical systems, numerical analysis, engineering mathematics, number theory, Riemann zeta function, and more. I don't think I will be rejected due to a lack of course experience.

I have some research experience but have not published any of my work. For my MS degree, I wrote a MS project about the computational methods of the Riemann zeta function. I know how to program and included both mathematics and computer code in my MS project. I cannot objectively measure the strength of my MS project (if length matters, it was near 100 pages long). I did not publish any of my work. In 2013, I also spent a couple of weeks doing independent research in astronomy. Astronomy is a side interest of mine and is something that I read about. Similar to my MS project, I didn't publish any of my work.

Concerns: What are realistic expectations for someone in my situation? Will graduate programs look negatively on my application because I didn't publish the work done in my MS project or astronomy research? Will they also see my five years of experience in industry as something negative in my PhD application? Since I plan to start graduate school in the Fall of 2018, I have plenty of time to plan and study more material (although, that probably isn't going to help me if I don't publish anything). I am not super concerned about my GPA (it is between a 3.4-3.5 for undergraduate study and 3.7-3.8 for graduate study). I have been looking at the rankings of PhD programs and have spent some time looking at advice for future graduate students.

Since I have roughly 1.5 years before I would start graduate school, I have been proactively learning new material. I have signed up for proof based mathematics courses (including abstract algebra and real analysis) so that I can strengthen my application. I'm not sure if taking these courses is necessary, although I hate to waste time at night doing unproductive activities or just reading math books on my own without any sort of credential.

I am in my late twenties and am very aware of my chances for an academic job (statistically near zero). I would like to work in industry after I graduate. I am not opposed to the possibility of becoming a professor, although it seems impossible given the supply of younger students and my background.

Questions:

1. Do I have a good chance of getting into graduate school in applied mathematics? How could I figure this out?

2. My primary reason for going to graduate school is because I want to learn more. Some of this could be accomplished on my own (by reading math books or taking online courses at MIT OpenCourseWare). I would like to dive much deeper than my MS project.

3. Do you think that my plan is insane given that I probably wouldn't end up as a professor?

The alternative is to stay in industry and work in software for the next 30+ years. I would like to try something different and would presumably be good at academic research.

Edited January 27, 2017 by Axion004