Part II

(6) Background. A good way to win a 100 yard dash
foot race is to start one foot from the finish line.
Otherwise, get a head start. Or, get an 'unfair
advantage'.

So, for the approach of 'mathematizing' engineering,
start with a solid undergraduate degree in pure and
applied math. Then get a Master's concentrating on
selected topics in pure and applied math.

E.g., even if you want to get a Ph.D. in computer
science, as an undergraduate, mostly just f'get
about the undergraduate computer science courses and
be a math major instead. The undergraduate computer
science material won't much help you in computer
science research, but the math can be an
overwhelmingly strong advantage.

As a math major, after the usual undergraduate
calculus sequence, take theorem proving courses in
abstract algebra and linear algebra -- the first is
useful at times, maybe more so in the future, and
the easy place to learn to prove theorems, and the
second is likely the most important material in
mathematical analysis and applied math.

To continue, take a second course in linear algebra.
A good start is the now classic P. Halmos, 'Finite
Dimensional Vector Spaces'. It was written in 1942
when Halmos, a fresh Ph.D. from Doob at U. IL, was
an assistant to von Neumann at the Institute in
Princeton and is really a finite dimensional
introduction to von Neumann's Hilbert space. For
more, see the books by R. Bellman or R. Horn.

Work carefully through W. Rudin, 'Principles of
Mathematical Analysis.' Supplement that with
Fleming, 'Functions of Several Variables', Buck,
'Advanced Calculus', etc.

As a crucial start on optimization, work carefully
through, say, Chvatal, 'Linear Programming'.

Somewhere get a good theorem proving course in
ordinary differential equations; that can be a good
start in a lot in applied math and on deterministic
optimal control theory.

At one of the first chances, take a careful pass
through Royden, 'Real Analysis' and the real half of
Rudin, 'Real and Complex Analysis' -- a good course
would help weight the more and less important ideas
there.

Then a long dessert buffet is Luenberger,
'Optimization by Vector Space Techniques'.

With that background, you can do probability and
stochastic processes the serious way via the more
serious authors, e.g., L. Breiman, D. Brillinger, K.
Chung, E. Cinlar, J. Doob, E. Dynkin, I. Karatzas,
M. Loeve, J. Neveu, S. Shreve, and more.

For more, attend research seminars: Don't often try
to follow much of the content but just use the
seminars as suggestions for new fields, problems,
techniques, names, references, etc. Also you may
meet some people or learn about career opportunities
that could be helpful.

If after a Bachelor's or Master's want to drop out
of school for a while, then do so, get a job, stay
single, and outside of work lead a simple life and
study some more pure and applied math from some of
the best sources. Hopefully get some practice
applying math to real problems.

Get a collection of real problems that might use for
your Ph.D. research (remember, your Ph.D. is to be
in engineering). Make some first cut progress on
those problems. If you can get some solutions, then
write them up as papers. If some of your papers
look publishable, then try to publish them.

(7) Ph.D. Program. Now apply to a Ph.D. program.
Include your papers, especially the published ones.

Your mathematical background can be a huge
advantage. In particular, consider working in
optimization and stochastic processes with a solid
background in measure theory and functional
analysis. Heck, nearly everything going on in a big
application, a server farm, a large network, ...,
the economy is a stochastic process where we want to
optimize.

Do what the department insists on, and otherwise
continue your research. Take your best research and
submit it as your Ph.D. dissertation. If there is
any question about the quality of that research,
then publish it; or just submit work that you have
already published. That the work was accepted for
publication in a good journal tends quickly to
settle all doubts about sufficient quality.

In particular, proceeding in your Ph.D. program as
outlined here, you have (A) obtained your own
background in pure and applied math, not depended on
your department for that background, and have quite
likely obtained a better background than any of your
professors; (B) have selected your own research
problems and not depended on your department or its
professors for research problems; (C) have done the
core of your research in applied math with theorems
and proofs which are comparatively easy to show are
correct and difficult to criticize; and (D) have
used publication of your work preemptively to
establish a respected, outside, objective proof of
quality.

(8) How to Do the Research. Mostly the mathematics
is just math and not very intuitive and is solid and
not just guessing. But finding that math, that is,
original math, can use quite a lot of intuition and
guessing for finding what might be true and finding
ways to prove it is true.

My view is that the most important work in math
research is intuitive with a lot of guessing, a lot
of simple models, and a lot of simple, intuitive,
testing of intuitive guesses.

(9) University. Generally you are better off at
the best research university you can get into. The
less good universities can force you to jump through
no end of silly hoops and be so insecure in their
own expertise as to delay and delay saying that your
work is good.

(10) Courses. At a really good university, it may
be that the graduate courses are not much like
undergraduate courses and, instead, are essentially
just introductions to narrow parts of research by
experts in those parts and, really, intended only
for students wishing to pursue research in those
parts.

(11) Done.

Accept your Ph.D. and go do something else.

Your professors may have helped you get a job, if so
likely an academic job; thank them, and if that job
is what you want, then take it; else proceed with
your career along lines you've had in mind.

(12) Warning.

A Ph.D. program can be dangerous, harmful to you and
your life and career and even fatal. If you are not
well protected with your own background in, say,
math, your own research problems, your own research,
and your own publications, and hopefully your own
financial means, then your education and much of
your life can be in the hands of others who can be
clumsy, competitive, nasty, arrogant, domineering,
abusive, destructive, sadistic, incompetent, etc.

It can be that, really, your professors don't have
any good research problems for you. E.g., even in
an engineering school, they may have nearly no
contact with real problems from off campus and,
thus, little or no help in finding a real problem
for your start. Your background in each of real
problems, math, computing, and business can easily
be much better than theirs. Each of your professors
may have been beating their head against some hard
problem for the last 15 years while you have some
good insight into some good, new problems you have a
good chance of solving. It is accepted that one of
the keys to success in research is good problem
selection.

You can feel that you are in jail without being
accused of a crime, with an indeterminate sentence,
tortured by your professors as jailers, and with no
chance of parole.

There is a special warning for students who made
Valedictorian in high school and PBK, 'Summa Cum
Laude', etc. in college. In the text version of D.
Knuth's 'The TeXBook' is:

"The traditional way is to put off all creative
aspects until the last part of graduate school. For
seventeen or more years, a student is taught
'examsmanship', then suddenly after passing enough
exams in graduate school he's told to do something
original."

So, a student who has done really well based mostly
on fantastic memory, pleasing the teachers, dotting
i's and crossing t's, doing just what was requested,
working desperately for praise and approval from
others, terrified of any chance of criticism, out to
'change the world' in major ways or bust, can find
themselves in a situation of inhuman stress, then
depression, then incapacitation, then more stress,
then clinical depression, then death. No joke.