Whenever I found a way to apply research-level mathematics to a business problem, the solution always was too difficult to sell. Sometimes I can solve my own problem, e.g. options pricing models known from financial mathematics can tell you the minimum to accept in a salary negotiation.
(2) Solid mechanics modeling, e.g., for jet engine turbine blades -- lots of math.
(3) Molecular spectroscopy: Some group theory and group representations can write down the possible spectral lines. That's long been the key to identifying molecules.
(4) My post mentioned Maxwell's equations for the design of the WWII radar that helped England win the Battle of Britain.
(5) A good, general purpose curve fitting technique is least squares spline fitting -- there's some math in there.
(6) Which pharmaceutical salesman goes to which physicians and leaves what samples to maximize revenue? Can be formulated as min-cost, capacitated network flows via the network simplex algorithm with a modification of W. Cunningham.
(7) Given a special scenario of global nuclear war limited to sea, estimate how long the US SSBN fleet would last.
(8) Given current revenue and planned capacity, project revenue growth -- solution saved FedEx.
(9) Given some ocean wave data, find the power spectrum, with confidence intervals, and generate sample paths with that power spectrum.
(10) At FedEx, which airplane goes to what cities in what order to carry all the freight, meet engineering, FAA, and safety constraints, arrive in specified time windows, and minimize costs.
(11) For an airplane, how to climb, cruise, and descend to arrive on time and minimize costs. Basically deterministic optimal control.
(12) Blue positions SSBNs and targets missiles in Red areas; Red attacks the Blue SSBNs; the remaining SSBNs fire. How best to position the SSBNs and target the missiles?
(13) A missile is chasing an airplane. What flight path should the airplane follow to evade the missile and what flight path should the missile follow to hit the airplane? A problem in differential games.
(14) A resource allocation problem in marketing resulted in a 0-1 integer linear program with 600,000 variables and 40,000 constraints. How to get a feasible solution optimal or nearly so? Solution involved non-linear duality theory and yielded a feasible solution bounding showed was within 0.025% of optimality in 900 seconds on a 90 MHz computer.
(15) My current startup.