Part I

I tried a grad math department and didn't
like it: (1) In a course in real
analysis, early on the prof discussed some
set theory. The summer before I'd had an
NSF thing in axiomatic set theory, Suppes,
von Neumann, an appendix in Kelley, etc.
His first test had a problem, and at the
last minute I saw a solution and wrote it
down. He called me on the carpet -- nasty
guy. I apologized for using little omega
for its usual meaning without defining it,
and then he saw that my solution was
better than his and I was off the
carpet. Bummer. He was too quick to
cut me off at the knees. (2) Course was
in Kelley, General Topology. As a ugrad
senior, I'd lectured a prof once a week
and covered all of it except the last
chapter on compactness when I cut out to
finish my honors paper [The typing was so
hard that from rolling the carriage a half
step my left arm hurt for a year!] But the
course in grad school, same book, was
beneath me. I turned in a stack of solved
exercises and was a nice guy -- I didn't
submit any I'd done in ugrad. Waste of
time. (3) There was an abstract algebra
course from Herstein's book -- by then
nearly all beneath me. E.g., my ugrad
honors paper had been on group
representation theory which is heavy
linear algebra and abstract algebra stuff.
I solved some exercise in ring theory and
got sent to a full prof. The only thing
new in the course for me was Galois
theory, so I studied that some weekend and
took an oral exam for the course. Waste
of time.

I wanted the math for math-physics but
didn't see how to get that there.
Certainly not Galois theory. There were
some good ways but not with the courses
they put me in. The specs for the q-exams
were a disaster -- the faculty committee
had a political train wreck. Bummer.

I got recruited by the NBS&T in DC.
Getting to DC then was the land of milk
and honey for applied math. I got
married, and she went for her Ph.D.

We had a great time, good French cheese,
some quite good French wine, lots of
plays, concerts, trips to Shenandoah, etc.

I got into descriptive statistics,
multi-variate statistics, statistical
hypothesis testing, numerical linear
algebra, curve fitting, the fast Fourier
transform, second order stationary
stochastic processes, extrapolation, and
power spectral estimation, optimization,
the Navier-Stokes equations, did a lot of
catch up reading in the basics, a lot more
in linear algebra, multi-variate calculus,
e.g., exterior algebra, and more. Kept
busy. Had a great time. Also got into
computing in a fairly big way. Got some
nice items, e.g., two new cars, etc.

My favorite book on my bookshelf,
including for applied math, is J. Neveu,
Mathematical Foundations of the Calculus
of Probability.

Worked in industry and saw some problems
in combinatorial optimization,
deterministic optimal control, and
stochastic optimal control, identified a
problem in stochastic optimal control and
found an intuitive solution, applied to
grad school in applied math. Got into
Cornell, Brown, Princeton, and more.

Independently in my first summer did the
research for my dissertation in stochastic
optimal control. Had lots of delays
having to do with my wife and, then, our
budgeting. In a rush, wrote some
corresponding software in two months, much
of it over Xmas at wife's family farm, and
wrote and typed in the final dissertation
in six weeks, stood for orals, and got my
Ph.D.

During Ph.D., did work in military systems
analysis, some optimization, statistics,
and Monte Carlo -- wrote the corresponding
software.

The day my wife got her Ph.D. she was in a
clinical depression from the stress. To
help her get better, I took a job I didn't
want as a B-school prof in applied math
(also played a leadership role in campus
computing and did some consulting) but was
near her home family farm that I hoped
would help her. It didn't. I took a job
in AI at IBM's Watson lab and did some
optimization, mathematical statistics, and
AI. My wife never recovered from her
illness and died.

Then I became an entrepreneur.

I did some interesting work in two cases
of optimization; thus I found good
solutions to the customers' problems that
they believed could not be solved; that I
solved the problems scared them off. One
solution turned out to be just linear
programming on networks -- I was coding up
the W. Cunningham variation when the
customer ran away. The other problem was
just some Lagrangian relaxation; I got a
feasible solution within 0.025% of
optimality in 500 primal-dual iterations
in 900 seconds on a slow PC to a problem
in 0-1 integer linear programming with
40,000 constraints and 600,000 variables
-- scared the pants off the two top people
in the customer's company. They had tried
simulated annealing, failed, and concluded
that no one could solve their problem;
that I found a good solution, both the
math and the software, scared them off.

I looked into lots of stuff that didn't
work out.

Lesson: US national security, especially
around DC, was, maybe still is, really
eager for a lot in applied math --
optimization, stochastic processes, etc.
In wildly strong contrast, I've seen no
interest in business at all comparable,
not even in Silicon Valley. The US DoD is
often quite good at exploiting applied
math; in comparison,
business, in a word, sucks. The
flip side of that suckage is, in some
cases, an opportunity.

Lesson: Business is still organized like
a Henry Ford factory where the supervisor
knows more and the subordinates are there
to add routine muscle to the thinking of
the supervisor. Sooooo, US business just
HATES anyone who knows more than any of
the supervisors about anything relevant to
the business, and one can about count all
the good cases of applied mathematics in
business without taking shoes off.

Business CAN make good use of specialized
expertise and does with lawyers, licensed
engineers, and medical doctors. Each of
these, however, is usually outside the
usual organization chart pecking order, is
often from an outside service, in a
research division, in a staff slot off the
C-suite, etc. Each of these has a
profession that is crucial; applied math
doesn't. Bummer.

In business, an applied mathematician who
shows the company how to save 15% of the
operating costs is a lose-lose to the
C-suite: If the project flops, then it
was a waste, and anyone in the C-suite who
signed off on the budget has a black mark.
If the project is successful, everyone in
the C-suite feels that their job is at
risk from the guy who did the good
project. So, the C-suite sees any such
project as a lose-lose situation.

Nearly no one in US business got promoted
for doing an applied math project
successfully or got fired for not trying
an applied math project.

So, sure, to make money with applied math,
go into business, your own business, as
your own CEO, and own the business.

Now some of the opportunities are closely
related to the Internet -- take in data,
manipulate the data with some applied
math, maybe somewhat original and novel,
spit out valuable results. Then
monetize the results whatever way looks
best. Use the math as a crucial, core,
powerful, technological advantage, secret
sauce. Don't expect the customers/users
to see anything about the math -- just get
them results they will like a lot. Do the
other usual things when can -- viral
growth, network effects, lock in, good
publicity, own data, etc.