A. In business, whenever you can find a use and start a company to pursue it.
Note:
(1) I know some math: My Ph.D. dissertation was on the applied math of stochastic optimal control. The work was advanced enough mathematically to address measurable selection, solved a real problem, and made some contributions to making such calculations faster.
(2) I'm pursuing a use of math in business. My current work is an information technology startup based on a Web site. The key to the promise of the work is some original math I did based on some advanced prerequisites.
So, I know something about applications of math.
Jobs? Early in my career, I knew some computing, was at GE which was shrinking (again) in computing, sent some resume copies, and in two weeks got seven interviews and five offers. The computing worked for a job.
Broadly the video is nonsense, contemptible, deceptive, misleading, badly supported nonsense. Students need to be warned.
Wall Street? In graduate school, Wall Street was one of the career directions I had in mind. So, I got a relatively good background in stochastic processes. Later I wrote Fisher Black (as in the Black-Scholes option pricing model) at Goldman Sachs and got a nice letter back saying that he saw no opportunities to apply math on Wall Street.
I interviewed in computing at Morgan Stanley, showed some nice applied math I'd done, and mentioned that I wanted to work on automatic trading but got no interest.
I got a call from Google, and they wanted only C++ programming.
I got a call from Microsoft and explained some work I'd done, and published, that would help Microsoft but never got past the first 'phone-screen' girl.
Early in my career, there was some interest in applied math, but this interest was essentially only for work paid for by the US DoD and for the Cold War. The US DoD has done a lot of good applied math, but those successes have had essentially zero influence in business.
Can look at the Web sites of 500+ US venture capital firms, nearly all in Silicon Valley, near Boston, or in New York City, look at their investment interests, and never see the words math or mathematics. Chris Sacca dropped out of a math Ph.D. program and may have the best background in math of any venture partner in the US. His biography may have the only mention of math on a venture firm Web site.
The interest of math in information technology venture capital is somewhere between zero and deeply negative. E.g., I got back from Ron Conway's son at Andreessen Horowitz
"your company is outside our area of expertise".
By "expertise" he had to mean the math, not the Web site.
Yes, in topics central to Hacker News, there should be interest in math for 'machine learning, data mining, big data, ad targeting' etc., but there is not. Instead, the world would rather stumble along with intuitive approaches to these topics instead of solid (powerful, valuable) math approaches.
Here's the main point: In business, the US is still organized mostly like the factory floor of 100 years ago where the supervisor knew more and the subordinate was there just to apply muscle to the ideas of the supervisor. And that 'inequality in technical knowledge' continues all the way to the CEO and COB who want to believe that they know the most.
So, no one in middle management wants to bet their budget or career record on some project using some math they don't understand; they don't get bad marks for such projects they do not pursue.
Essentially only a CEO could sponsor a project with some significant math, and CEOs rarely sponsor projects directly.
In venture funded information technology entrepreneurship, the 'technology' is supposed to be just routine software that could be understood by a bright, self-taught middle school hacker and, thus, can safely be ignored as crucial 'intellectual property' by any Ivy League history major, Harvard MBA venture partner.
Broadly, bringing high technical expertise as an employee to a business is a fool's errand. Yes, there are a few exceptions, e.g., some technical topic deep in the CIO's organization, but generally the employee is supposed to know less, not more.
An employee who, on the side or whatever, starts a good project based on math will find that their management doesn't like the project. If the employee proposes the project more broadly, then they will be told not to communicate outside of their management chain or face getting fired.
Once as a professor, hired in optimization, I wrote a research paper in optimization based on some math, with theorems and proofs. My department reviewed the paper and concluded that it was "not publishable". It published right away, without significant revision in the 'Journal of Optimization Theory and Applications'. At Yorktown Heights, we were doing some work in server farm and network monitoring of health and wellness using artificial intelligence. I found a much more powerful approach with some original work using some advanced topics in stochastic processes, and the lab concluded that the work was "not publishable". The work published right away without significant revision in 'Information Sciences'. The reason: The people I was working for didn't know math and didn't like it.
Really, if math is to have a significant role in a business, then the main mathematician has to be the founder, CEO of the business, e.g., Jim Simons, Andrew Viterbi. Otherwise, f'get about it.
Broadly, math is by far the most exacting and difficult academic subject, is rarely pursued by students beyond what is needed for K-12 teaching, physics, engineering, or basic statistics, and is just not known, liked, wanted, or even tolerated in society more broadly.
For a good career, mostly an electrician's license is a better foundation than a Ph.D. in math, electrical engineering, or anything else.
My view is that math is by a wide margin the most powerful and valuable topic for the future of information technology entrepreneurship for at least the first half of this century, but to make this promise real the mathematician has to be the founder, CEO of the company, and the math will not be an advantage in seeking equity funding.
To make this promise understood, about the best I can do is be successful financially, buy a yacht over 200' long, and give some interviews on the yacht cruising in Long Island Sound.
Broadly our society does not value, want, like, or tolerate math.