
Math Overflow users resolve PhD thesis crisis - DarkContinent
https://mathoverflow.net/questions/366765/issue-update-in-graph-theory-different-definitions-of-edge-crossing-numbers
======
wenc
This reminds me of my Ph.D. crisis. (which I'm sure many former grad students
can relate to)

I was in my 6th year. All my friends had graduated, and my stipend had run
out. I was 2 weeks away from submission and discovered that one of my
assumptions was wrong, which potentially distorted/invalidated all my studies
-- to fix these studies would have potentially delayed submission for months.
It was a very subtle assumption violation (and it wasn't even that wrong) and
my committee probably wouldn't even have noticed. I was tempted to sweep it
under the carpet and not let it keep me from graduating.

But I knew it was wrong. I felt that if I sacrificed my integrity then, the
moral failure would mark me for life. No one would know -- but I would know.
So I decided to fix the issue, re-do the studies and live with the reality
that I would have to delay my defense.

Turns out when you're desperate -- and many grad students can attest to this
-- a resourcefulness that you never thought you had kicks in ("where were you
during all my years of grad school?"). I don't remember how, but I somehow
managed to wrangle new studies out in 3 days (which would have previously
taken me months). I made the deadline in the end.

The lesson I learned was that committing to doing the right thing has its
costs, but in some cases it also forces one to explore attacks never
previously considered. Asking on MathOverflow is one such attack.

~~~
supernova87a
There's another (slightly kinder) version of this:

When people go to grad school, then during their studentship get married and
have a kid -- suddenly their productivity goes up dramatically and output per
time increases by many factors. What happened?

It turns out when you have self-motivating reasons to get out (and a target on
your head from your spouse to earn some money for godssake), you find ways to
focus on what's important and drop the rest.

No more idling away for hours on silly ideas that don't get you closer to
handing in your thesis. No more trying random libraries that get your code to
run 2% faster. No more goofing around after 6pm with other students just
because you have the time -- you have to get home and be a breadwinner for
your family. You have to get shit done.

You start to ask, "even if I don't know exactly what the thesis will say, how
should it be organized and what _kinds_ of conclusions will make up the
writing? And what experiments do I need to fill in those charts/paragraphs,
and _no more_?" What's the minimum I need to do to get out of here? Not, "What
amazing interesting thing could I explore?"

Limits and constraints sometimes free the mind dramatically. The side effect
is maybe you don't get to explore ideas that go nowhere, but that's a
discussion about the purpose of the PhD and for another topic I guess.

(And sometimes, if you think, well I don't have a kid, so what's the rush?
Well, someday you might have a spouse, a kid, and every day of time you left
in grad school is a day for your future self -- and family -- and $$ -- left
behind in time. Work to free your future self... now, while you have the
time.)

~~~
JamesBarney
Hmm this is different than my experience when my co workers had kids. Usually
their productivity dropped due to being sleep deprived and not having the time
on nights and weekends to come up to speed on any new technology.

Not that I minded, just saying that time constraints and sleep deprivation
seemed to have the effect you'd except from them.

~~~
iateanapple
> Usually their productivity dropped due to being sleep deprived

High levels of sleep deprivation is only common for the first few months.

> not having the time on nights and weekends to come up to speed on any new
> technology.

This tends to be a problem on teams that don’t properly evaluate costs of new
technology and so churn like crazy for very minor productivity increases.

------
oconnor663
A neat comment on the accepted answer:

> From OP's point of view this could be viewed as glass half-full rather than
> glass half-empty. Their dissertation results hold unequivocally on the
> sphere and might hold on the torus, though it is an open problem if they do.
> It is certainly legitimate to study what follows from a given conjecture
> being true. It could even be spun as a feature rather than a bug of the
> dissertation. If the results in fact fail on the torus then you know that
> the conjecture must be false. Potentially, it could open up a fruitful
> avenue of attack.

Kind of reminds me of Terence Tao's post on what solving big problems looks
like: [https://terrytao.wordpress.com/career-advice/be-sceptical-
of...](https://terrytao.wordpress.com/career-advice/be-sceptical-of-your-own-
work/)

~~~
thechao
When Terence Tao writes stuff like this, I'm always very happy that I got to
experience the Moore-method for learning math (at UT Austin). A group of us
would be dumped into a class with a common topic and we'd just have to _prove_
things (topology, algebra, analysis) ... on the blackboard, in front of
everyone. The best work we did was when something started going wrong and then
we'd all start arguing about the proof, building count-conjectures on the fly
and riffing on the math. The worst work was when someone went and found a
proof _ahead of time_ and just showed the answer. There's so much to learning
where the sharp bits of math are; proofs are the razor-thin path through the
briar patch.

It was only later that I found out that history, the study of art &
literature, and philosophy can all teach you the same thing. The important
part is that you're interested in the topic.

~~~
nicoburns
> > There's so much to learning where the sharp bits of math are; proofs are
> the razor-thin path through the briar patch.

As a student representative for my undergraduate mathematics course, I got
really pissed off at lecturers for exactly this reason: they'd write out a
perfect correct proof on the whiteboard, but wouldn't explain where it had
come from or how people had arrived at the solution. We were left to figure
that out on our own.

They then complained that students were rote-learning for exams, rather than
coming to a full understanding of the material. I'm not sure what they were
expecting, given that that's exactly how they were teaching it.

~~~
nwallin
My discrete mathematics professor was like that. He would regurgitate a proof
onto the whiteboard. Then he'd do it a few more times with proofs of other
things.

He has an identical twin brother, who is also a math professor at the same
college. The regular professor was out for a day, and his brother came in to
teach the class. His teaching style was completely different. "Ok, we need to
prove X. Where should we start?" and would sit on the table and look at us
with an inquisitive look on his face. Then learning happened.

Everyone's mind was blown. Most people didn't realize it was a different
person. Then on Thursday it was back to same-old same-old.

I still don't know shit about discrete math.

~~~
benibela
I am not getting anywhere with the latter approach when TAing

I ask them such a question, and then wait for 15 minutes, and no one says
anything.

------
lifeisstillgood
The walled garden sweet spot

Stack overflow and it's cousin sites have many serendipities like this - and I
happily conjecture this happens more here than facebook or twitter.

I think the reason is that despite being a walled garden (ie proprietary) it
still has a promise to open up the content and makes effort to moderate and
grow the community - in other words what they are really selling is not the
SEO but the sweet spot between "anyone posts anything" of an "ideal" internet
where no rentiers exist but no one can find anyone else, and the much more
corporate hand of Facebook.

I am not sure reddit exists in this sweet spot either - mostly because there
is just sooo much reddit.

~~~
Shog9
A couple relevant bits of info about MathOverflow:

\- The site is operated by Stack Overflow/Exchange, but is _owned_ by
MathOverflow, Inc a non-profit corporation[0]. As such, it retains the right
to exist independently of the Stack Overflow company - to my knowledge, it is
the only public Stack Exchange site for which this is true.

\- Like all public Stack Exchange sites, authors retain ownership of their
work, which is published under a CC-BY-SA license. Regular archives are
uploaded to Archive.org and can be obtained there or via Bittorrent[1]

In short, not a walled garden, and not Stack Overflow's garden.

[0]: [https://meta.mathoverflow.net/questions/969/who-owns-
mathove...](https://meta.mathoverflow.net/questions/969/who-owns-mathoverflow)

[1]:
[https://archive.org/details/stackexchange](https://archive.org/details/stackexchange)

~~~
lifeisstillgood
Re: MathOverflow Inc - I had not only not heard of this, but never even
considered it was possible :-)

Yes, I think I am wrong to use the term walled garden, but it's hard to think
of something else.

In a "platonic ideal" of the internet everyone would have their own internet
connection, and a server and say post their own interesting queries and
somehow others would find and answer them.

Perhaps search was assumed to solve it all then.

But the universe is much more "clumpy" than that so people will gather around
certain locations, in nature they are natural oasis.

Perhaps we should drop the walled garden idea - gardens, walled or otherwise
need tending and upkeep and that passed the ability of one or two people to do
in their spare time somewhere around 1991 on usenet.

Tending a garden is a costly affair.

I think perhaps walled city might be a better term? It implies the "never
leaving" which is what facebook seems to aspire to.

perhaps a better analogy is "chargeable car parking". :-)

~~~
Shog9
This is yet another one of those situations where the use of a dying
metaphor[0] hurts communication; you wall up a garden to protect what is
inside from the harsh conditions outside: wind, cold, vermin... The
implication is that the people in the garden are delicate flowers who would be
destroyed by the conditions on The Greater Internet if they were to be
exposed.

The antonym to the walled garden is the open garden or field, with hardy
plants able to withstand and even thrive in the local atmosphere. They're
still _cultivated_ \- weeded, fertilized - but there's no need to create a
microclimate to just to accommodate them.

In this context, Facebook does make some effort toward walling off their
gardens, but... As you note, Facebook's primary goal isn't protecting it's
_dominating_ \- Facebook is just as happy to own major portions of the 'Net in
pursuit of this goal, and more than a little reluctant to provide any real
protection beyond what is absolutely necessary.

Beyond that... We all garden. From little personal websites and blogs, to big
community gardens[1] like Wikipedia, Reddit, Stack Overflow, and even Hacker
News. We plant, we harvest, we tend these plots, alone or together, but make
little effort to isolate them from the larger world - indeed, we generally
recognize that the strength of the Web is based on its interconnections, its
inherent ability to draw together different sources of information.

[0]:
[https://www.orwell.ru/library/essays/politics/english/e_poli...](https://www.orwell.ru/library/essays/politics/english/e_polit)

[1]: [https://meta.stackexchange.com/questions/349513/feedback-
for...](https://meta.stackexchange.com/questions/349513/feedback-for-the-loop-
june-2020-defining-the-stack-community)

~~~
lifeisstillgood
Interesting - my understanding of the walled garden was the Omar Khayyam style
of a luscious oasis walled off to allow only a few people to enjoy it whilst
keeping most out (the implication that you had to pay to be one of the few).

Neither definition actually makes much sense when talking about incompatible
protocols.

~~~
ngcc_hk
One of the major difference when you think of any concrete items like flower,
gardens is that they are private goods. Public goods especially information,
laws of physics, idea, mathematics etc. are hard to create but no costs to
consume. Hence, the walled garden is not about keeping most people out of view
i.e. Wall Street J etc. but keeping the noise from the creation so there is a
point for the OP to post his question in a panic. Public goods like F=ma is
very hard to be created and what motivation to create one is a big question.
This is especially if it is networked public goods. And some public goods are
useful to one or one group but not the others. That is the fundamental problem
here.

Guess the one above this poster has right that the analogy is not totally
right. Still if one assume the flower can be viewed a trillion time but the
question is creation, walled garden is a good metaphor.

~~~
lifeisstillgood
The first usage I remember of walled garden in an online sense was about AOL
and their refusal to use common internet protocols - you literally could not
email from outside AOL and if I remember not view websites outside AOL. There
was a wall around their garden. IIRC it became a common description in tech
columns of papers.

I still see the walled garden analogy to be more useful

If people are confusing their definition of a metaphor and mine (I don't think
Orwell mentioned that but maybe it's low down the list of stylistic errors)
then that can cause problems. But I struggle to see how Facebook / twitter
"protects" creators from the harsh winds of the outside.

~~~
Shog9
This is where the "dying metaphor" thing comes into play: AOL was a walled
garden in the sense that I outlined - like so many other early online services
(and BBSs), it had walls to protect its members. That was the original sense
of the metaphor.

But... That sense is dying; it is a poor metaphor _because_ few people
actually build real-world walled gardens[0][1] for that purpose; the metaphor
has no currency, and thus the meaning shifts. Now it is just as frequently
used to describe any sort of system which restricts the flow of people OR of
information, in or out. So while Instagram might be considered a "walled
garden" in the original sense (no outbound links on posts), it may ALSO be
considered a "walled garden" in the sense that it restricts inbound access for
non-members, or even in the sense that it forces certain onerous licensing
terms on contributors. In this manner, the metaphor becomes problematic, as
what one intends to convey is not necessarily the meaning which is understood
by others. If/when the metaphor dies entirely, becoming an idiomatic way of
saying "not completely open", this problem disappears - no one will attempt to
relate people to plants, or content to flowers.

With this in mind, I suspect your original intent was focused on the "garden"
aspect: that the value MO provides comes from imposing a structure and certain
expectations which facilitate productive interactions like this, with the
"sweet spot" being that it remains _open enough_ for the rest of us to benefit
from the outcome of those interactions.

[0]: [https://solar.lowtechmagazine.com/2015/12/fruit-walls-
urban-...](https://solar.lowtechmagazine.com/2015/12/fruit-walls-urban-
farming.html)

[1]:
[https://news.ycombinator.com/item?id=22395292](https://news.ycombinator.com/item?id=22395292)

------
generationP
As far as notational clusterfucks go, crossing numbers (along with the three
standard definitions of a ring) are one of the best-known ones to still be
biting people on a regular basis. ("Positive" and "natural number" are
sufficiently well-known that people are careful.) But imagine how it felt to
do group theory back when "group" could mean any of "abstract group",
"subgroup of GL(n)", "finite group", "monoid", "semigroup" and combinations
thereof.

~~~
tgb
The simplest gotcha I know is: is f(x) = 1/x piece-wise continuous?. This is
calculus 1 level material and yet author's disagree significantly on this
point, sometimes without specifying it! Some say yes, others would require f
to have finite left and right limits at every point. This mattered for a point
of my thesis and my advisor was very unhappy with me calling these function
piece-wise continuous.

~~~
generationP
I thought this was only an issue in K-12, as anyone in research math considers
the domain and the target to be part of a function, and then the problem
disappears: The function R \ {0} -> R, x |-> 1/x is not just piecewise
continuous but continuous on-the-nose. The function R -> R, x |-> 1/x doesn't
exist. The function R |-> some completion of R, x |-> 1/x is continuous or not
depending on which completion you choose (the one with two infinities or the
projective line).

But I do recall a similar confusion happening with "piecewise linear" (the
question is whether the pieces have to fit together).

~~~
tgb
No you've missed the distinction. In all cases the domain and range are R (you
can fill in a value at 0, it doesn't matter which). See the MathWorld page
which leaves the definition intentionally ambiguous:

"A function or curve is piecewise continuous if it is continuous on all but a
finite number of points at which certain matching conditions are _sometimes_
required."

Emphasis added.

[https://mathworld.wolfram.com/PiecewiseContinuous.html](https://mathworld.wolfram.com/PiecewiseContinuous.html)

~~~
generationP
I don't get this. If you require f to be piecewise continuous outside of a
finite set of points _and_ to satisfy left limit = right limit at each of
these points, then you just have a continuous function. Why another word for
it?

~~~
tgb
The left and right limits are required to exist (and finite), but not
necessarily to be equal to each other. So f(x) = 1/x and sin(1/x) are out but
x/abs(x) is not.

------
fizixer
OP is incredibly fortunate. Or maybe mathoverflow is that active/supportive.

As a STEM grad student (not in math), I had more than a couple such moments of
crises, when I posted my questions on various stackexchange websites. I got
either useless replies, or no replies.

~~~
iflp
Mathoverflow is different from most of the other SE sites in that it's only
for research level questions. There is a separate site, math.stackexchange,
for other math-related questions.

------
supernova87a
I have to say also that this type of crisis is not surprising (unfortunately)
for math, or similar highly theoretical, _loner_ fields. I can guess that the
student asking the question is not being very open with his/her advisor, has
worked and struggled for long hours alone, thinking they have to solve it on
their own, and is not super communicative and checking in about important
aspects of the thesis. Because he/she thinks it has to be a surprise
"breakthrough" result -- a heavy obligation of the field's expectations.

No responsible advisor would let the work get to such a state, so late in the
game. Major fault of the advisor too, here.

~~~
jfkebwjsbx
Advisors are also very much at fault, not just students.

The last year of my PhD I ended up being pretty much alone because my advisor
had changed research topics a year before and therefore was not interested nor
up to date, so any of her inputs were not very useful.

A couple friends of mine also struggled with their advisor because he actively
avoided communication for some reason. I guess he had a personal or health
issue.

So even on good faith, advisors can end up making students life quite
stressful for one reason or another

------
thaumasiotes
I was contacted by someone in a PhD thesis crisis who wanted me to provide
speech samples they were apparently missing. The thesis was due imminently.

As far as I could tell, the analysis was already done -- but my samples were
needed for some other reason. I was kind of bemused by the idea that the
analysis would be invalid with nothing behind it, but valid with unrelated
data behind it.

------
ggm
My insight was an eng. student whose novel outcome of a maths model in Fortran
on a mainframe depended on his not understanding what uninitialised arrays
were. This was in the 80s.

There was no interesting novel outcome: he was random-sampling prior states of
memory.

I felt very bad for him, it was mid-stage. I didn't hear how he resolved it.

The other side of this is the crisis which only emerges in the viva. I was
working in Leeds uni in the 80s and overheard an external discussing a case he
had: it was obvious the results were fraud. They made the student and his
supervisor to the sums in the room, on the blackboard. He didn't get his
thesis.

------
benibela
I was in a PhD crisis, but I did not post it anywhere. Not sure if it is
allowed to ask for outside help.

Although now I have finished the thesis without that part (it should have
become an additional chapter). Perhaps I should post it around (although that
might spoil it for a paper)

Consider n polynomial equations in variables x_1, .., x_n, with constants
a_1,..,a_n, b_1,..,b_n, c_1,...,c_n:

    
    
         p_1 := a_1 x_1 x_2 + b_1 x_1 + c_1 x_2 + d_1 = 0
         ...
         p_{n-1} := a_{n-1} x_{n-1} x_n + b_{n-1} x_{n-1} + c_{n-1} x_n + d_n = 0
         p_n     :=   a_{n} x_{n} x_1 + b_{n} x_{n} + c_{n} x_1 + d_n = 0
    

Under which circumstances exists a (unique) solution for x_1,..,x_n in terms
of the constants?

I have found a recursive approach that results in a quadratic equation,
containing only a single variable x_i (and the constants). (It is too much for
a comment, here is a PDF: [http://benibela.de/tmp/quadratic-equations-
recursion.pdf](http://benibela.de/tmp/quadratic-equations-recursion.pdf) )

For example for n = 2, it is very simple: x^2_1 (a_2 b_1 - a_1 c_2) + x_1 (a_2
d_1 + b_2 b_1 - a_1 d_2 - c_1 c_2) + b_2 d_1 - c_1 d_2

This gives 2 solution. But I do not know what happens if the terms cancel each
other out. Like if a_2 b_1 - a_1 c_2 = 0, there would only be one solution.
But since the full solution in the pdf is so complex, I do not see which
constraints would lead to cancellation there.

\---

And that is not the full problem I was trying to solve. In the full problem
there are constraints on the a, b, c, d. There is a given graph, and depending
which nodes are connected in the graph, the constants are the same. Like if
node 3 and node 7 are connected, then b_3 = c_7 und c_7 = b_3. (even more
complex though). And then the question is, do these constants cancel in the
solution of those equations? And the final problem we want to solve: which
graphs lead to exactly one solution, and which graphs lead to no solution of
the equations?

------
no_identd
>"the faculty will not accept asymptotics"

the hell does that mean?

~~~
impendia
(Math professor here).

Ambiguous and poorly explained. (Note the question immediately afterwards
asking for clarification.) But probably something along the general lines of
"My advisor said that, if my main theorem is an asymptotic estimate instead of
an exact formula, then this would not be judged to be novel/strong enough to
earn a Ph.D."

~~~
mywittyname
If you don't mind, could you explain the practical difference or the reasoning
behind such a requirement? Are there situations where boundaries appear to be
asymptotic but the exact solution shows this not to be the case?

~~~
rrobukef
I re-implemented a quasi-polynomial algorithm. Experimentally, it shows
exponential behaviour. Back-of-the-envelope calculation shows this behaviour
can continue until the input size is >>10^21 before the asymptotic bound
asserts itself. (For comparison, input size 30 is unfeasible)

~~~
Ragib_Zaman
Which algorithm?

~~~
rrobukef
Parys' quasi-polynomial algorithm for solving parity games,
[https://arxiv.org/pdf/1904.12446.pdf](https://arxiv.org/pdf/1904.12446.pdf)

Note, another implementation doesn't have this behaviour for the family of
inputs I use. It's an implementation detail that has no effect on correctness.
Thus for the other implementation another family should exist.

------
rvieira
I'm totally sympathetic with feeling the clock ticking for a PhD thesis
submission (been there), however:

I knew I could take this route but never did it. This is a bit of a mockery of
the whole purpose of a PhD thesis (it has to be _your_ work primarily).

This is just an advanced version of posting homework questions on the
internet.

------
jb775
Wisdom of the crowd in action.

I feel like this wisdom isn't tapped into enough. We're often burdened with
individual tasks and challenges while utilizing crowd knowledge is looked down
upon or seen as an inferior solution finding mechanism. e.g. Imagine if
companies worked together to figure out self-driving cars rather than compete?

~~~
alentist
There’s the Autonomous Vehicle Computing Consortium (AVCC):
[https://www.avcconsortium.org/aboutus](https://www.avcconsortium.org/aboutus)

------
thom
This seems like a deeply flawed qualification, that smart people, noticing
interesting properties of deep problems, are forced to panic in such ways.

~~~
omginternets
On the surface, I agree: there's an interesting problem that's worth solving,
and a purely artificial limit is forcing people to do a bang-up job at solving
it.

But if you dig a bit deeper, I can see two counter-arguments:

1\. The _real_ risk -- by which I mean "the risk I have most often observed in
the wild" \-- is that a Ph.D expands to fill the time it's given, without ever
wrapping up and producing a publishable result. This happens so often that
it's practically expected in some places.

2\. Having a deadline, oddly enough, also serves as a catalyst for birthing an
idea... for "pinching it off" as the expression goes. At some point you have
to stop planning and start executing. You can see the deadline as a forcing
function.

Ph.Ds are needlessly traumatic and procedural in many ways, but I'm no longer
sure that hard deadlines are a net negative.

------
ReedJessen
This kind of "social media" gives me faith in the ability of humanity to
survive and thrive in the future.

------
blickentwapft
You don’t often see mathematics emergencies.

------
leokennis
Low quality comment, but this once again confirms what I already knew: I suck
at math.

