Part II

If you do much with computer graphics you
will encounter matrix theory. That takes
you into linear algebra; next to
calculus it is likely the most useful
math. Evidence: There are a lot of
downloads of LINPACK.

Can start a course in linear algebra by
considering solving several equations in
several unknowns. The standard technique
is Gauss elimination, and can program
that in about one page of code. Linear
algebra is a good start on curve fitting
in statistics and the math of quantum
mechanics.

If you want to understand more about
cryptography and error correcting codes,
you should study abstract algebra. Here
I would suggest that you actually take a
course (a) to help you get through that
quite different world of thought and (b)
especially to learn how to write proofs.
And for (b), take a course where the prof
is really good and also carefully reads
and comments on your proofs. Abstract
algebra is the easy place to learn to
write proofs.

Can get more guidance on how to learn math
at

https://news.ycombinator.com/item?id=28215105

Somehow long, maybe still, knowledge of
both math and computing can be welcome and
lucrative in parts of US national
security. That was the case early in my
career when my annual salary was 6+ times
the cost of a new high end Camaro.

Soon FedEx had what their founder, COB,
CEO called their "most important problem"
-- fleet scheduling. The BoD was
concerned, and crucial funding was at
risk. I typed furiously, wrote some
software, the output "solved" the problem,
enabled the funding, and saved FedEx.
There, sure, needed to calculate great
circle distances so used the law of
cosines for spherical triangles -- solid
geometry can be good stuff! Also had to
handle wind vectors -- linear algebra can
be powerful stuff. Then I went off to do
much more, integer linear programming set
covering where can discover much of the
motivation for currently the most
important problem in computer science, P
versus NP.

Later the BoD wanted some revenue
projections. I did a little with some
calculus and got a nice answer. Long
story short, that work saved FedEx a
second time.

For another long story -- I needed to be
better at office politics -- I just missed
out on some FedEx stock that should be
worth ~$500 million now.

The US Navy was collecting ocean wave data
at sea, and I was in a software house
bidding on writing some software to
analyze the data. One customer engineer
wanted (a) to know the power spectrum of
the ocean waves (that is, what
frequencies have the power) and, then,
(b) to generate synthetic, random
ocean waves with that power spectrum. I
quickly read a book by Blackman and Tukey,
typed in some software, showed the
engineer the results on how to find the
power spectra (with an important point
about handling low frequencies) and how to
generate the synthetic waves, and our
company got "sole source" on the software
work.

Later at IBM's Watson research lab, we
were doing AI for monitoring of server
farms and networks. I thought of another
way, for some of the monitoring much more
powerful than the AI, based on some
original math, and published the results.

Net, some math, especially through
calculus and linear algebra, can at times
be an important career advantage. For
more, get good with probability theory, if
you can, the version based on the subject
measure theory. Then learn some about
stochastic processes. E.g., once the US
Navy wanted an evaluation of the
survivability of the US SSBN (missile
firing submarines) fleet under a special
scenario of global nuclear war limited to
sea -- in two weeks. From some old work
by B. Koopman, I saw a continuous time,
discrete state space Markov process
subordinated to a Poisson process, wrote
some software, and was done on time. My
work got reviewed by a famous
mathematician, and he questioned how my
software could "fathom the enormous state
space". I answered, at each time, the
number of SSBNs surviving is a real valued
random variable. It is positive and not
greater than the number of submarines to
begin with so is bounded and has an
expectation and a finite variance. Then
the law of large numbers applies. So,
generate 500 independent sample paths,
average them, and get the expectation
"within a gnat's ass nearly all the time".
He agreed. I passed the review!

If you go for a Ph.D., then understand
that, in the US, academic positions at the
better universities are about three
things, research, research, and research,
especially because that leads to grant
money. The operational definition of
research is that it got published in a
peer-reviewed journal. If you publish,
say, 3 papers a year, then likely people
will stay off your case and you will
likely make progress to tenure. People
making the promotion and/or funding
decisions will rarely look at the papers
and, instead, just count them. Papers
that result in prizes are usually quite
powerful for a career. Generally, though,
academics is not very promising for
providing a good standard of living and
good financial security for you and your
family and these days can't hope to
compete with what is available in
computing, the Internet, etc.

Then the math? It can be an advantage.
The "advantage" can have you push ahead,
maybe by a little or a lot, useful
technology, economic productivity, and
civilization. Such progress happens,
actually fairly regularly, but is rarely
easy. So, if want to push civilization
ahead, (a) don't expect that the work will
be easy but (b) math can be one of the
most powerful advantages.

Now you know some of what I wish I'd known
at the beginning of my career. I want a
do-over -- where can I apply?