Have you applied to any programs where the degree is actually "Applied Mathematics"? This comment may apply to the program you list too though. Often you can still do "pure" research in such a program, proving theorems, etc. You could pursue some theory in probability theory/stochastic processes or statistics and the related.

It sounds like you might have a fixed project you are assigned to though. Without knowing the details of that project, it's hard to say what your options really will be. Do you think you can put up with working on that project for 4+ years? Even if you end up not liking the project, liking the people you work with might make up for some of that. But it might really just depend on your own personality/disposition.

For future employability, the information theory and machine learning is probably the better choice. Of course, it is difficult to predict what the economic and political situation will be when you finish. As long as humanity still exists within a technologically advanced civilisation, what you learn will likely still be quite valuable.

On that note, consider the job prospects of previous graduates. Where do they typically go? Programs often keep a running list of their graduates. Maybe there are existing pipelines to private industry or public sector.

It could also be helpful to look at current job advertisements that sound interesting to you to see their desired applicant qualifications. Take note especially if any computer skills listed. Continue to scan such job adverts every now and then, updating your self-study program on the side if/as needed.

Getting a faculty job on this field will also be more lucrative than math. Your starting salary will be a significant amount above that for a starting math assistant professor. This of course will vary quite a bit depending on your accomplishments and where you apply. But, as mentioned elsewhere, a greater proportion of PhDs in such an applied field will be going to industry, but those academic programs are facing higher and higher demand, so basic economics works in your favor if you want to be faculty in an engineering department or similar.

It's a lot to consider, but it's an important decision. Make sure you can live with your choice, or, at least, if you decide to change course, be honest with yourself about it and take appropriate decisive action.

Good luck!

Have you applied to any programs where the degree is actually "Applied Mathematics"? This comment may apply to the program you list too though. Often you can still do "pure" research in such a program, proving theorems, etc. You could pursue some theory in probability theory/stochastic processes or statistics and the related.

It sounds like you might have a fixed project you are assigned to though. Without knowing the details of that project, it's hard to say what your options really will be. Do you think you can put up with working on that project for 4+ years? Even if you end up not liking the project, liking the people you work with might make up for some of that. But it might really just depend on your own personality/disposition.

For future employability, the information theory and machine learning is probably the better choice. Of course, it is difficult to predict what the economic and political situation will be when you finish. As long as humanity still exists within a technologically advanced civilisation, what you learn will likely still be quite valuable.

On that note, consider the job prospects of previous graduates. Where do they typically go? Programs often keep a running list of their graduates. Maybe there are existing pipelines to private industry or public sector.

It could also be helpful to look at current job advertisements that sound interesting to you to see their desired applicant qualifications. Take note especially of any computer skills listed. Continue to scan such job adverts every now and then, updating your self-study program on the side if/as needed.

Getting a faculty job in this field will also be more lucrative than math. Your starting salary will be a significant amount above that for a starting math assistant professor. This of course will vary quite a bit depending on your accomplishments and where you apply. But, as mentioned elsewhere, a greater proportion of PhDs in such an applied field will be going to industry, but those academic programs are facing higher and higher demand, so basic economics works in your favor if you want to be faculty in an engineering department or similar.

It's a lot to consider, but it's an important decision. Make sure you can live with your choice, or, at least, if you decide to change course, be honest with yourself about it and take appropriate decisive action.

Good luck!

For what it's worth, I chose applied mathematics and probability theory and am pleased with that choice since the ability to teach statistics is an in demand skill too. I didn't have the computer skills necessary for industry jobs though. That's why I emphasize that latter aspect. I'm tenured faculty in an undergraduate math department. That also brings up another piece to mention: consider the fact that there might be smaller colleges to apply to for faculty positions too. Maybe you will easily qualify for R1, but consider honing your teaching skills too just in case. Not that teaching skills aren't important at an R1, but they will be more important when applying to teaching-focused jobs such as a small liberal arts college (say, one with an engineering or math major).