Well, for nearly any
topic in applied math, there
are some really simple
treatments; they might be
fun to read, but they are like
a bicycle -- don't want to use
them to cross the Rockies.

Instead, often want the real
stuff. For that, usually need
a good version of the math
prerequisites or will struggle.

Where I had good prerequisites,
I did fine. Otherwise I struggled.

The graduate work that did me
the most good was some quite serious work in optimization,
measure theory, functional
analysis, probability theory,
and stochastic processes.

So, then, presto, bingo, look
at Fourier theory, L^1, L^2,
etc. and prefer it to the
best ice cream and cake. Gorgeous. Powerful. Great
fun -- and I understand it.
Before the grad studies,
I was never quite sure
what the heck had a Fourier
transform and what didn't,
when I could interchange order
of integration, etc. It was
like driving a car but not knowing
how many wheels it had.
At one point I got pushed
hard into the fast Fourier
transform, and taken narrowly
that was okay, but the narrow
view is not the one really want,
and I had to struggle with the
broader view. E.g., if have
the power spectrum of the noise
and that of the signal, what
filter do you want? Generally
if do know what linear filter want, then can use the FFT to
implement it. At one point
it would have been good
to have done such things,
but I didn't
know how. Now I'm sure I could
read it, likely from N. Weiner and then from more recent sources and find it easy reading
and/or just derive it myself.

Just why is the power spectrum
the Fourier transform of the
auto-covariance or some such
result? Then I didn't quite
have the background for that;
now I'm tempted just to
derive it myself.

Ergodic theory? It was also
great fun. Before the grad
school work, ergodic was
just a mystery.

Now I can browse and thoroughly understand and enjoy Luenberger, Optimization
by Vector Space Techniques,
e.g., Kalman filtering, deterministic optimal control,
high end versions of Lagrange
multipliers. A lot of
it is based on the Hahn-Banach
theorem, and I saw a rock
solid version of that. Before
grad school, Luenberger
would have been a bit much -- I would have struggled, often
not quite sure just what I
was doing, limited to
some painting by the numbers.

Before grad school, at times I
wanted to know Kalman filtering
and deterministic optimal
control but struggled. It would
have helped for me to have
known those topics.

I wrote my dissertation in
stochastic optimal control,
but I wished I'd been able to
have a good, high end course
in such things, including with stochastic differential equations, etc.,
but there wasn't much in
courses to pick
from. The number of US
departments that know and
teach that stuff is tiny.
Off and on I considered
just some independent reading
courses for that material,
but various exogenous events
caused me to run short on time
and cash and rush to finish instead.

Before the grad work, I
found statistics to be
a cookbook of bad tasting
meals.

Now the usual statistics books
bore me; I don't know of a
really good statistics book;
and I usually end up just
deriving what I need for
myself. With a good background
in probability, that's the way
to go.

I had a pretty good undergrad
major in math, and when that
was enough as prerequisites
I did well. E.g., I touched
on linear algebra as an ugrad
but wanted more. So, I got
a stack of books on LA and
dug in. Of course, the best
was Halmos, Finite Dimensional
Vector Spaces, but that didn't cover everything closely related,
e.g., applications to calculus of several variables, numerical
linear algebra, connections with multi-variate statistics, etc., and I got books on those for more.

My ugrad work didn't do really
well with multi-variable calculus -- no wonder because
too soon really need measure
theory. So, I got partly
caught up on such calculus
on my own and got a lot more
in grad school, but
I still don't know
differential geometry
well enough to find general
relativity easy and wish I did.
At least now I have the
prerequisites to learn
differential geometry. E.g.,
as an ugrad, the courses never
covered the inverse and implicit
function theorems (just local
nonlinear versions of what is
obvious in the general case of
solving linear equations), but
I got some good treatments
in my independent reading.

E.g., in computing, sure,
Bachus-Naur form was easy enough,
really is basically just
set theory, but I never
got how to take such BNF and
automatically write a parser.
Once it might have been
nice to have done that! Or just
use Yacc, right?

> Ph.D. requisite?

A good Ph.D. is a good thing.
It's the training in how
to work with things that are new.

So, learn to zip through a lot
of stuff don't need to know
well, where K-12 and college
and even grad courses have
an implicit norm that such
zip work is not good. Good?
Heck, it's crucial!

Learn the pros/cons of
stuff that is new.
Do begin to lose patience
with things that are old --
too often they are not as
good as might assume in a
course and should be improved
on or even just set aside.

In doing research, just have
to look at the material
in a more effective way than
the usual way as a student
trying to make good grades.
No longer trying to make good
grades and, instead, are trying
to do something new, e.g., improve
on the stuff in the text books
or papers.

Also are no longer hanging on
every word of a prof and, instead,
are trying to do own stuff.

Some of the coursework for
a Ph.D. is also usually
darned good. At a good school,
i.e., a top research university,
the difference in quality is
like that between
a dinner at McDonald's and
one at a Michelin 3 star --
no joke. Or, if have a
course in ugrad school and
think that the course was
fine, if get a course from
a good researcher, very bright,
who knows the material from
various approaches, can get
a LOT more. The good
researchers are smarter --
and not by just a little bit.

My view is that math, and
mostly the advanced stuff
and often new stuff,
will be the key to the
future of computing, at least
until the software is able to
do math better than humans.
Computing only what we
can think of without math
will become way too limiting.