Stresses of a Ph.D. program? Yup.

(1) My wife was high school Valedictorian
and in college Summa Cum Laude, Phi Beta
Kappa, and Woodrow Wilson and NSF Fellow.

For her Ph.D., she did fine in her
coursework. But starting research, she
struggled terribly: For her, the research
itself was easy. Her problem was that,
unlike coursework, what was good, good
enough, too good, what was really wanted,
the realities of research, were totally
unclear leaving her terrified of doing
something wrong.

The stuff of professors being not helpful,
giving poor advice, finally being too
critical was real and devastating.

She struggled on with the anxiety, right,
causing depression. That actually hurt
her speed for which she got criticism, the
first ever in her academic work, and more
stress and depression and finally real
clinical depression. She was in severe
clinical depression the day she got her
Ph.D. She never recovered. She was
visiting her family farm trying to
recover, and she was missing. Her body
was found in a lake.

Yes, the OP is correct: Grad school can
be a very destructive place.

(2) As a high school senior, I took the
SAT twice and got over 750 both times. At
my high end, college preparatory, high
school, of 1, 2, 3 on the Math SAT, I was
#2. So, eventually it became clear I had
some math talent.

Well, I figured out I had some math talent
in the 9th grade -- I ignored the class
and taught myself from the book and then
made As on the tests. I continued that
way in high school math classes.

The extreme case was plane geometry -- I
loved the subject. The teacher was the
nastiest teacher I ever had, so I slept in
her class and refused to admit doing any
homework. But in fact, I solved every
nontrivial problem in the text, including
the more difficult supplementary ones. On
the state test, I was #2 in the class; #1
was the guy who later beat me by a little
on the SAT Math.

I was a horrible student. Why? In grade
school, my brother was ahead of me and a
good student. The teachers kept saying,
"Oh you are Joe's [not his real name]
brother!". Well, from that start,
anything I could do would be lower than
expectations. Moreover, I was a boy; all
the teachers were women; I had no social
insight on how to please those women; my
handwriting was awful; ...; and I just
gave up trying to please the teachers.
The teachers talked to each other so much
that I was labeled as a bad student with a
new teacher before I even started her
class. That's why I was a horrible
student -- in grade school, for me to
please the teachers seemed just hopeless.

Well, that poor student approach continued
in college: I nearly ignored every
subject but math and physics, was a math
major, largely taught myself, did some
more advanced reading independently
outside of class, took a reading course
from a high end, pure math text where I
did all the talking, lectures and
exercises, and a prof just listened, and
did write a math honors paper.

Then I worked for a while in applied math
and computing on US national security
problems and later at FedEx. Learned a
lot of math, job related and also just
good pure math.

Then I went to grad school. I already
knew about enough for the qualifying
exams.

But I still preferred to work directly
from some of the best texts. Class time
was mostly just writing exercise copying
from the board. The real learning was
studying the text or class notes.

Eventually I saw a problem not solved in
class; I didn't find a solution in various
relevant texts or the more famous relevant
papers in the library so took a "reading
course" to investigate, maybe just write
an expository paper, on the problem.
Well, it was a fun problem. I had some
ideas and worked them out sitting with my
wife on our bed while she watched TV.
Some of my ideas were good, and I turned
them into solid theorems and proofs. In
two weeks, I was done. Fast reading
course! The credit I got was the last I
needed for an MS.

But it was clear that what I'd done was
publishable. Later I did publish in JOTA.
One prof walked up to me in the hall and
said "I heard about your theorem: It says
that for Brownian motion ...." Right.
So, with that work, original, publishable,
done with no faculty direction, presto,
bingo, I had a halo.

Before grad school, I'd seen a problem.
On an airplane flight I worked out an
intuitive solution. Some coursework in my
first grad school year gave me enough pure
math background to turn my intuitive
solution into solid math, and I did that
independently in my first summer; walked
out of the library with a 50 page
manuscript. That was the original
research for my dissertation. Later I
wrote some illustrative software, polished
my math, and that was it: There was no
real faculty direction. I stood for an
oral exam defense and got my Ph.D.

Lessons:

(1) For a Ph.D., the three most important
things are research, research, and
research. If get some good research done
early on, then can have a halo and solid
shield against criticism.

(2) It helps a lot to be able to be a poor
student in courses. This way you are used
to criticism and don't take it seriously.
Also you don't wear yourself out trying to
get a Ph.D. just from working hard in the
courses or pleasing the profs teaching the
courses. Bluntly, can't get a Ph.D. just
from doing well in courses. Indeed, at
the high end grad school I went to,
officially there was no coursework
requirement for a Ph.D. Instead, the main
requirement was just research -- "An
original contribution to knowledge worthy
of publication.". So, (A) I didn't burn
myself out doing coursework. If I didn't
like a course or thought it wouldn't do
much for me, then I didn't do much in the
course or just walked out. (B) But I DID
work hard learning the good material, and
for that worked much as I had, largely
independently, as I'd done back to the
9th grade.

(3) Work independently. Don't ask or
expect the faculty to do much or anything
to suggest problems, references, or
approaches to solutions. Find your own
problem, maybe even before going to grad
school. Start research on your problem
ASAP, in some sense in your first year or
first summer. Before you and the school
agree you are working on your
dissertation, have it essentially all
done, all but maybe some illustrative
software and some polishing of the work,
e.g., maybe more references.

(4) For financial support, expect free
tuition but no stipend. Have your own
source of food, clothing, shelter, etc.

(5) Stay close to math so that don't have
to do lab work for some prof or for your
research.

(6) Pick a practical problem, get a good
applied math solution building on mostly
some pure math, and call the work
engineering. The usual criteria for
research are that the work be "new,
correct, and significant." Well, can get
the "significant" part from the importance
of the real problem. Can get the "new"
part from the particular math, somewhat
new, for the new solution to the new
particular practical problem. The new,
practical problem can stimulate and
justify some new math that otherwise might
not be regarded as of value or interest.
For "correct", have the core of the work
theorems and proofs which are seen to be
correct.

So, where my wife wore herself out trying
to get praise from the profs, I was
willing to be a poor student and largely
f'get about trying to get praise. Where
my wife wore herself out trying to wonder
if her research was good enough, I tried
to do good research as I understood it and
also that looked good enough for the
criterion of "worthy of publication".
Looking for praise, my wife was reluctant
to do something original. I was ready to
do something original whether I would get
praise or not. Indeed, back in plane
geometry, I'd taken a problem from outside
of class, found a solution, after school
showed that nasty teacher, and she said
"You can't do that". Later I discovered
that she was wrong and I'd reinvented a
classic technique similitude and shown
her. I didn't believe her. If are going
to work with things that are new, realize
that will likely get a lot of people,
hopefully not quite everyone, saying that
the work is not good.

What I did can sound like Ph.D.
hacking. Well, not really: IIRC, at
least at one time the Web site of the
Princeton math department just stated:
(1) No courses are given for preparation
for the qualifying exams. Students are
expected to prepare for the qualifying
exams on their own. (2) Courses are
introductions to research taught by
experts in their fields. (3) Students
should have some research work in progress
as soon as possible, hopefully in their
first year.

Well, that's basically what I did: I was
already nearly prepared for the qualifying
exams before I got there. I had a
research problem started before I got
there. The courses I worked hard on were
good for my background and for my
research. The course where I got the
problem that led to my paper in JOTA was
advanced enough that I could see a good
unsolved problem.