The Mathematical Hacker (2012)
I was a &quot;programmer&quot;, and well paid at the time, but also had a good background in ugrad plus some in pure&#x2F;applied math.<p>The math helped, was enough help to be a crucial difference and put me way ahead of the pack, that is, everyone else around.  The math I did was to solve problems in the job I had.  The job was as a programmer or other computer guy title, but I saw the problems and used math to attack them successfully.<p>So much for some generalizations.  Some examples are needed:<p>Example 1:
One afternoon Fred Smith, founder, COB, CEO of FedEx, stumbled out of his office, tired, frustrated saying &quot;we need a computer&quot;.  He had just been trying to schedule his fleet of airplanes.  Soon the BoD was also concerned, so concerned that crucial, necessary equity funding was seriously at risk.<p>A guy I knew in ugrad physic class called me.  At the time I was being a computer guy at Georgetown U. and also teaching courses in computer science. Several of us met in a conference room in the library to discuss what to do about the scheduling.  There was lots of noise. Finally I announced that I would design and write the software.  I got an account on a time sharing service offering CP67&#x2F;CMS (VM&#x2F;CMS) computing and wrote the software in my favorite language, then and now, PL&#x2F;I.<p>Later in Memphis, one evening SVP Roger Frock and I used my software to develop and print out a schedule for the full planned fleet.  The next day two representatives of Board Member General Dynamics went over the schedule and announced &quot;It&#x27;s a little tight in a few places, but it&#x27;s flyable.&quot;  The BoD was happy, and the funding came.  Smith&#x27;s remark was that the schedule &quot;solved the most important problem&quot; facing the company.<p>Role for math?  Apparently I was the only one around who could see that calculations of great circle paths was just the the law of cosines for spherical triangles and to do the vector calculations to handle winds.<p>Right, not much math was involved, but the math was crucial, and I was the only one around who had it.<p>Example 2:  At Georgetown a computer science prof got some public code for some statistical operations, wrote a main program to call that code, and used the result in teaching a course in statistics.  In his testing, three of the public routines had problems.  Two of the problems I fixed with just some PL&#x2F;I tricks with memory and some algorithms from Knuth.  For the third, there were numerical problems, and I solved those with a version of orthogonal polynomials.  Not much math, but the math solved the problems the prof saw, and I was the only one around who knew that math.<p>Example 3:  I was in a software house, part of KMS (early connection with laser fusion), and bidding on some software the Navy wanted.  Part of the work was Nyquist sampling, the fast Fourier transform (FFT), power spectral estimation, digital filtering, etc.  I had been working close to all that due to working with the FFT, got the Blackman and Tukey book on the statistics of power spectral estimation, quickly wrote some sample code, in PL&#x2F;I, illustrating how to do much of what the Navy wanted in the bid, showed the code and its output to one of the Navy engineers, and presto, bingo KMS got coveted &quot;sole source&quot; from the Navy.  My work with the math put our software house far ahead of the software houses we were competing with.<p>Example 4:  The BoD at FedEx wanted some revenue projections.  People around the office had hopes, intentions, dreams, etc. but nothing rational or convincing.  We knew the current revenue and the revenue from the full, planned fleet.  So, the projections were an interpolation between those two.  Argue that the revenue would grow by current customers influencing customers to be so that the rate of growth would be proportional to the number of current customers doing the influencing and the number of customers to be being influenced.  So, at time t, let the revenue ($ per day) be y(t), the current time t = 0, the current revenue y(0), and the planned revenue b.  Then for some constant of proportionality k, we should have<p>d&#x2F;dt y(t) = y&#x27;(t) = k y(t)(b - y(t))<p>So, this is a first order, linear, ordinary differential equation initial value problem.  There is a closed solution based on, right, exponentials.  It is just calculus to solve the equation.  I did that, picked a reasonable k, drew a graph, and, ..., in short this work kept the BoD from <i>collapsing</i> and saved the company.<p>Calculus.  Gee, just calculus.  My big advantage was that I remembered calculus (quite well, and way beyond this problem) and was able to formulate the problem as calculus and solve the differential equation.<p>Right, the solution is a <i>lazy S curve</i> and is related to some models of Covid growth.<p>There were other examples in 0-1 integer linear programming, some original work in mathematical statistics, continuous time, discrete state space Markov processes, etc.<p>Here is how that can go:  Are working in computing in a company, see a problem the company has, see some math that provides a solution, and, by writing some software and using available data, proceed with some success.  Maybe no one else in the company sees the problem because they don&#x27;t know enough math to see a math formulation and solution.<p>It appears that commonly where people are doing a lot of important work, there are important problems that need solving, are being neglected, and where some math can be the key tool in formulation and solution.  A person with that math may be unique in the company.<p>But don&#x27;t hold your breath looking for job descriptions that mention such usages of math.  Uh, why not?  Did I mention that the person with the math might be unique in that organization.<p>Also draw from the two old Disney movies <i>Snow White</i> and <i>Cinderella</i> and see the possible roles of jealousy and sabotage.  In my experience, Disney was 100% correct.