> You’ve told us vague things about wide areas of mathematics, but you haven’t told us what problem your start-up is solving using these branches of mathematics or how it is a better allocation of investor dollars than focusing on ML.

Easy: ML is quite narrow. More options yield the same, usually better, investment results. Compared with the applied math on the shelves of the libraries, ML is nearly pinpoint narrow. In addition, too much of ML is heuristic curve fitting short on math guarantees. Also, where some real problems have some features that justify some math assumptions, ML is slow to exploit those.

In the workshop, kitchen, or lab, use the best tools you can. So don't limit yourself to just what saw on a cooking show on TV.

My view is that mathematical proof is a fantastically powerful part of applying math. The core reason is that it lets do some additional, maybe significantly original, derivations to get better solutions for real problems.

E.g., once I was in an AI group working on monitoring and managing server farms and networks. We got rivers of data on many variables at high data rates. There we wanted to detect problems ASAP in near real time.

Okay, necessarily, inescapably, we have two ways to be wrong -- false positives (claiming the system is sick when it's healthy) and false negatives (claiming the systems is healthy when it's sick).

Okay, then, right away we are forced, like it or not, into the framework of some statistical hypothesis tests continually applied. There, as is usual
in hypothesis testing, we want to be able to select a false alarm rate and get that rate exactly. We'd like the most powerful test as in the lemma of Neyman-Pearson but likely do not have enough data. Still we want test power in some good sense; if we are willing to tolerate a higher false alarm rate, then we'd like a higher detection rate; when we get a detection, we'd like to know, as a measure of seriousness, the lowest false alarm rate at which the data would still be a detection; at least we don't want a trivial test.

For this data, no way can we know the probability distribution of the data under the null hypothesis (the system is healthy). This is because even the univariate data is commonly a total mess, far worse than was long common in statistics (U. Grenander explained this to me in his office at Brown). And, of course, with so much data on so many variables, we should be doing multi-variate testing.

So, net we need tests that are both multi-variate and distribution-free. Go look for those in E. Lehmann, etc. and won't find any.

So, I invented some. Then, how the heck to adjust false alarm rate? That was a problem in applied probability that needed new theorems and proofs. So, I did those. For those I used measure theory, measure preserving from ergodic theory, group theory from abstract algebra, and more. Yup, theorems and proofs.

Otherwise I would not have known what the heck I had, what the properties were, what the false alarm rates were, etc. Then do my tests have any power? I didn't have data enough to apply Neyman-Pearson, but I found a way. Are my tests trivial? I used a result of S. Ulam and stated and proved a theorem -- not trivial.

The work was intended to be practical. Likely and apparently the work remains the cat's meow, best in the world, for detection of problems never seen before -- behavioral monitoring, zero day monitoring.

Gee, just read that the last security attack on Target cost them $300 million. They needed better monitoring!

New theorems and proofs were crucial, and the work was intended to be practical.

A guy had a fast opportunity to do some marketing but, with the short time interval, had limited resources. He had a 0-1 linear programming formulation, 40,000 constraints and 600,000 variables for a small, test case. Sure, no linear programming or integer linear programming package would have a chance.

So, I got his formulation, saw some special structure to exploit, did some non-linear duality derivations for some Lagrangian relaxation, found how to get some bounds, wrote some code, and in 905 seconds on a slow computer found feasible solutions guaranteed to be within 0.025% of optimality. Well, the derivations were essentially theorems and proofs. He'd tried a computer science approach, simulated annealing. He ran for days and then stopped with the best he had and with no idea how close he was to optimality. My work, from theorems and proofs, was much, MUCH better.

The crucial core applied math of my startup is some theorems and proofs; I wouldn't believe that the mathematical operations would yield what I want otherwise.

Sure, usually a working applied mathematician doesn't always have to stir up new theorems and proofs, but at times that's darned helpful and valuable.