Still in academics, if want to make a big splash outside math departments, then math can be one of your best tools and advantages. One approach: Learn some measure theory and functional analysis, standard early math grad school topics. Then, less standard, learn probability, stochastic processes, and statistics based on measure theory. Learn some differential equations -- big part of math. Then, also less standard, learn some optimization and control theory, both deterministic and stochastic. Yes, I'm not nearly the first to suggest such math topics; in recent years they have been proposed as 'the mathematical sciences' (that didn't catch on nearly as well as hoped). Then use this knowledge to build best possible, 'optimal', 'models' that attack the ubiquitous 'uncertainty' in other fields, write papers, teach seminars and then courses, write text books, do consulting, get grants and grad students, etc. Be a prof, maybe, in finance or production in a B-school or in EE in an engineering school.
Math Jobs Outside Academics
Likely the biggest opportunity for 'jobs' in math outside of academics is the US Federal Government, especially with DoD funding, related to national security.
Otherwise, for a 'job' with any very significant role for math, f'get about it. Why? To have such a job, except in very small companies or the DoD path just above, someone needs to understand the work of the job, write a job description, get the job funded in their budget, and put some of their career on the line that the money will be seen by the more senior managers as money well spent. That is, in essentially all larger organizations, the ideas of the factory floor 100 years ago are still in place: The supervisor knows more than the subordinate, and the subordinate is there mostly just to apply more blood and sweat to the work of the supervisor. So, since nearly no supervisors know much math, f'get about such jobs.
Or, suppose there is a mathematician in a large organization. At the top there is the CEO who forgot any calculus they might have learned. Between the two is middle management. So, by a standard math argument, somewhere in that management chain must be a mathematician reporting to a middle manager non-mathematician, and that won't work. So, yes, maybe a mathematician can be 'on staff' to the CEO. Don't hold your breath.
But how do other technical fields such as law and medicine work? From licensing, malpractice threats, professional codes of conduct, professional practice peer review, they have a LOT of professional status -- math doesn't. Also they are applied fields with their graduate education aimed almost entirely at practice -- that is, is 'professional' training -- instead of research, etc.; math isn't like that. In particular, law has a standard that a lawyer can report only to another lawyer.
There's a LOT of advanced math in high end academic EE, but it remains that an electrician's license can be a much better foundation for a career.
The Main Opportunity
Outside of academics and government, a relatively stable career nearly always needs a relatively stable collection of happy, paying customers.
To skip to the bottom line, math can be an advantage if the mathematician owns the business that is, except for the math, much like other businesses from Main Street to Silicon Valley to Wall Street.
So, the mathematician uses the math to construct the crucial, core, powerful, defensible (difficult to duplicate or equal) 'secret sauce' and implements it in software that delivers valuable results. It is the results, essentially only the results, that the happy customers pay for.
Back to my claim of more important than Moore's Law: We already know what the world wants in the famous one word answer, "More!". The main way for more is automation. For that, so far we've been just coding what we already knew how to do by hand or just intuitive or heuristic ideas. The main way to get more powerful software (that is, able to generate more valuable results) is to have it implement more powerful manipulations, and the main way to that is math, yes, complete with theorems and proofs (so that we can have confidence in the work), possibly original based on advanced material. My view is that for the rest of this century, (1) this math direction is (thanks heavily to DoD projects of the past 70 years) well proven and rock solid, (2) progress better than via math is not promising, and (3) progress without the math is not promising. Of course, just now, one advantage is that nearly no one understands the math or accepts this claim!
For the academic math departments, their 'teaching' pyramid is at risk. To get their field going again, they need to 'connect with reality' and deliver value that plenty of other people are willing to pay for, hopefully quite directly, otherwise at least indirectly. "The analytic-algebraic topology of the locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some pursuit of abstract beauty no one else can appreciate are NOT good directions.
Money for Academic Math Applications
Still in academics, if want to make a big splash outside math departments, then math can be one of your best tools and advantages. One approach: Learn some measure theory and functional analysis, standard early math grad school topics. Then, less standard, learn probability, stochastic processes, and statistics based on measure theory. Learn some differential equations -- big part of math. Then, also less standard, learn some optimization and control theory, both deterministic and stochastic. Yes, I'm not nearly the first to suggest such math topics; in recent years they have been proposed as 'the mathematical sciences' (that didn't catch on nearly as well as hoped). Then use this knowledge to build best possible, 'optimal', 'models' that attack the ubiquitous 'uncertainty' in other fields, write papers, teach seminars and then courses, write text books, do consulting, get grants and grad students, etc. Be a prof, maybe, in finance or production in a B-school or in EE in an engineering school.
Math Jobs Outside Academics
Likely the biggest opportunity for 'jobs' in math outside of academics is the US Federal Government, especially with DoD funding, related to national security.
Otherwise, for a 'job' with any very significant role for math, f'get about it. Why? To have such a job, except in very small companies or the DoD path just above, someone needs to understand the work of the job, write a job description, get the job funded in their budget, and put some of their career on the line that the money will be seen by the more senior managers as money well spent. That is, in essentially all larger organizations, the ideas of the factory floor 100 years ago are still in place: The supervisor knows more than the subordinate, and the subordinate is there mostly just to apply more blood and sweat to the work of the supervisor. So, since nearly no supervisors know much math, f'get about such jobs.
Or, suppose there is a mathematician in a large organization. At the top there is the CEO who forgot any calculus they might have learned. Between the two is middle management. So, by a standard math argument, somewhere in that management chain must be a mathematician reporting to a middle manager non-mathematician, and that won't work. So, yes, maybe a mathematician can be 'on staff' to the CEO. Don't hold your breath.
But how do other technical fields such as law and medicine work? From licensing, malpractice threats, professional codes of conduct, professional practice peer review, they have a LOT of professional status -- math doesn't. Also they are applied fields with their graduate education aimed almost entirely at practice -- that is, is 'professional' training -- instead of research, etc.; math isn't like that. In particular, law has a standard that a lawyer can report only to another lawyer.
There's a LOT of advanced math in high end academic EE, but it remains that an electrician's license can be a much better foundation for a career.
The Main Opportunity
Outside of academics and government, a relatively stable career nearly always needs a relatively stable collection of happy, paying customers.
To skip to the bottom line, math can be an advantage if the mathematician owns the business that is, except for the math, much like other businesses from Main Street to Silicon Valley to Wall Street.
So, the mathematician uses the math to construct the crucial, core, powerful, defensible (difficult to duplicate or equal) 'secret sauce' and implements it in software that delivers valuable results. It is the results, essentially only the results, that the happy customers pay for.
Back to my claim of more important than Moore's Law: We already know what the world wants in the famous one word answer, "More!". The main way for more is automation. For that, so far we've been just coding what we already knew how to do by hand or just intuitive or heuristic ideas. The main way to get more powerful software (that is, able to generate more valuable results) is to have it implement more powerful manipulations, and the main way to that is math, yes, complete with theorems and proofs (so that we can have confidence in the work), possibly original based on advanced material. My view is that for the rest of this century, (1) this math direction is (thanks heavily to DoD projects of the past 70 years) well proven and rock solid, (2) progress better than via math is not promising, and (3) progress without the math is not promising. Of course, just now, one advantage is that nearly no one understands the math or accepts this claim!
For the academic math departments, their 'teaching' pyramid is at risk. To get their field going again, they need to 'connect with reality' and deliver value that plenty of other people are willing to pay for, hopefully quite directly, otherwise at least indirectly. "The analytic-algebraic topology of the locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some pursuit of abstract beauty no one else can appreciate are NOT good directions.