text:
How to stop hopping the learning chain and actually begin somewhere?
Here is what I feel whenever I find something interesting and feel like pursuing it:

Oh so I like X (Computer Graphics), let me read up papers/books about it.
Ok let me begin with reading up Y (OpenGL)
But Y needs W (Linear Algebra)
Well reading up Z (Probability) first makes more sense.
Umm, you shouldn't jump to Z without learning U (Permutation/Combination).
And how come I forget about reading V (Number theory)
And what not..

I always end up searching and reading up "Best books to begin A/B/C/D.." instead of actually making myself begin somewhere. This consumes all my energy and I never really start.
Q. Have others faced this? Q. How do you handle this and actually begin somewhere?
Any help would be really appreciated.
Background: I am a working professional, with a Masters in Computer Science (fascinated with Computer Graphics etc). It's been two years since my masters but I still kind of miss academia, my thesis work and other interesting stuff I did there. My current work is also pretty interesting and partially overlaps with my interest areas. However, other than work, I would really want to continue doing things related to my masters side by side (and MAYBE take up a PhD somewhere down the line). But the never ending feeling of not knowing anything takes over.
It's not that I am being forced to study any of this. It's purely for my personal interests that I want to pursue it.

Update 1:
Thank you so much everyone for such overwhelming response. I really like several answers and points raised but haven't marked any yet because I am in the process of trying some and would update if something actually works for me. Meanwhile, I also found this talk really helpful:
Inside the mind of a master procrastinator

Update 2 [2-Oct-2024]:
A couple years after this post, I did end up joining a part-time PhD program while continuing my full-time job. Unfortunately, after 5 years into it, I had to drop out, with not much progress made. I faced this issue even during my PhD and in general with life. At times I feel this issue stems from some unresolved mental issues I have. Just adding another link that makes sense at this point:
Dealing with Stress | Jeff Bezos

"But the never ending feeling of not knowing anything takes over." That is something you need to get used to in a PhD, so you already have a good qualification. ;-)

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Dan's answer is very good, and I want to add one more point:
Accept that on the first pass you will not be able to fully understand the topic. Appreciate that there are decades and sometimes centuries (for the math, at least) of research supporting the topic you're learning. Thinking you can "get to the bottom" of it in a few months, let alone a week or two, is not realistic. Choose a topic and try to understand that as much as possible. Keep notes on other interesting topics along the way, but don't get distracted by them; just accept them as assumptions as this point and move on. You can get back to them later.

I think this is exactly right.  Normally when I read a textbook for the first time I only expect to get a high-level understanding of the subject.  I'll usually skim large parts of it.  If it's a good textbook, maybe a year or two down the road I'll revisit it more carefully and do a few of the exercises.

Good answer, but you should remove the phrase "on the first pass".

The first pass is spot on imo. Since my teens I do a speed reading to any book from cover to cover, and then only I read it at a slower pace to try to understand something

@JeffE I disagree. The first pass is what is important here. Further passes will improve your understanding. You can't guarantee that someone won't fully understand the topic after a few thousand passes - and have wasted many years studying for a PhD.

If anyone could fully understand any subject, there wouldn't be research. That's what research is -- finding something that nobody understands, and understanding it better. There is always more to learn.

@JeffE - Your point is entirely correct. I'm leaving the phrase in as it emphasizes the point which newcomers need to hear, they will not learn it all on the first pass. The fact that they won't learn it all on their second or third or nth pass either is true, but (to me) is a secondary message.

I am just finishing my undergrad and I learned this lesson the hard way.

Naive question here: where is dan's answer ? I can't seem to find it...

@VincentFourmond - Dan is "user24098" now.

If you are actually new to a topic then the papers on it are probably out of your reach.
When I want to open a truly new topic I would start with the lowest level textbook I can get. Perhaps I find that a first year undergrad textbook is not challenging enough to be interesting, but its worth checking each chapter to see if there is anything new. If there is new content then it will definitely  be possible to understand without any prerequisites. Then I move on to a slightly higher level textbook, etc.
This should bring you smoothly up to the level of papers, and any periphery knowledge (from other subjects) needed will probably be presented at a level that is easy to access.
(full disclosure, I have only ever done this with topics in maths, perhaps it doesn't work well elsewhere)

First sentence is very important.  For the rest... textbook may or may not be the best, depending on personal learning style -- the important thing is starting with introductory material.  And practicing.

*defiantly --> definitely

This doesn't seem to address OP's problem, though. I think OP's problem is not so much in reading papers vs textbooks, or starting with something high-level vs low-level, but rather that on reading anything of interest he'll find more interesting things on a lower level and move to those. Even if he were to start with the lowest level textbook he can find, he'll later find that there are even lower level topics and move to studying those by his own habit.

@jlmg sure, but I get the impression the reason OP wants to read lower level topics because they feel that that will help with the higher level topic. My point is that provided you start with textbooks aimed at undergrads you wont need to have studied the lower level topic because the relevant bits will be presented as you need them.

Create something and learn in the process
If I want to learn a new programming language or framework, maybe I'll do a quick tutorial but then I immediately have to find something useful and/or fun to do with it. There's no end to the knowledge that you can get on any topic, but if you don't do something with it, at some point you'll just stop. If you create a need for yourself to learn in order to create something specific, chances are you'll get farther. It also serves as a stop gap, when you have enough knowledge to achieve your goal, you continue working on the project until you get stuck again.
Maybe this is less suitable for very theoretical fields but certainly it works for practical fields like computer science, certainly computer graphics.

Precisely this. I have 101 abandonned half-finnished projects on my hard drive, but I can explain how each and every one of them works.

My research is shifting from systems computer science (programming languages, compilers, performance engineering) to theory, and this approach is working great there too.

Spot on. I'd like to compare this with "agile" program development (doing something useful/tangible quickly) versus "waterfall" - waiting to churn out great output in one go

I totally experienced the same thing both in academia (MSEE) and work life, especially early on:

Life is not an academic knowledge competition, so no need to feel inadequate. There will always be people smarter or more knowledgeable than you in any specific topic/niche (see 6. below), with more letters after their name. Go for breadth-first, not depth-first (as @NateEldredge said).
Be guided by learning the minimum you need to know to get a specific task done. Always have a specific application in mind, and skip everything that's not relevant to achieving it (Be strategically lazy, in a goal-directed sense). If you don't have a goal, you will just go in circles. You can always bookmark the paper/documentation/textbook and come back and read it later as/ when/ ever you need to know more. Don't try to memorize anything you don't need to, which is 99.99% of what you will ever read.
Papers: Textbooks can be better than papers. Webpages can be better than papers. Online tutorials can be better than papers. Webinars, whitepapers, sales demos (the good ones, at least) can be better than papers. Going to a Meetup or talking to a smart friend over a drink can be better than papers. Often, papers are a posturing exercise in showing how clever, name-dropping or connected the writer is (or even bragging about their funding or the size of their servers), not in delivering concise timely accessible information to a lay audience (and I say all this as someone who published papers in my MSEE, and read/skimmed at least 10,000 papers from multiple disciplines). Even when the writer is genuinely attempting to get to the point, they may still not be as good at communicating as the other channels I mention above. If I read a paper and the author starts throwing dense jargon at me for no particular reason and citing bookshelves of erudite publications, I ask myself whether it's worth my while to dig into it, or try to find a dummy's notes, or even poll Google/citeseer/Google Scholar for the wisdom-of-the-crowd on whether a certain technology/algorithm/API/whatever is useful, relevant, applicable and competitive.
Ten 5-minute lightning talks or poster sessions can be better than one 1hr talk. Often, information density is proportional to log(duration). Commercial conferences can be better than academic ones (sometimes). Hallway conversations can be better than lectures.
Use the right resource. Mailing-lists, user forums and asking your question at a Meetup event are all great ways to source a quick wisdom-of-the-crowd opinion. As long as you show you've done your basic homework first and you're not a help vampire. (Conversely, give back, too, whenever you can)
Find strategies to deal with information overload/impostor syndrome. In general, become aware of and reject all of that, inadequacy, guilt, grief, existential angst, information overload for not reading all the footnotes and citations and wading through the proofs line-by-line. Information is being generated orders of magnitude faster than we can ever process in our lifetime, and accelerating. All we can do is filter the ever-diminishing sliver we need to get on with life. "I will always choose a lazy person for a difficult job. Because, he will find an easy way to do it" - Bill Gates. Personally I know one guy who trained an ML suite to automatically summarize the 10,000s of pages of quarterly technical reports he had to stay on top of in his field (IT security). He's a brilliant guy. This wasn't even in his job description. Always try to cut corners, automate, generalize, abstract, use third-party open-source libraries.
Friendships, professional acquaintances and contacts matter. Get to know smart people and shun useless people ('lazy' != 'useless'). Work your network. Never eat lunch alone (TM). And so on. Now you can tap on 1000 people all distilling information and experience. Again, don't overuse it, give back as much as you take, and keep a running note of what you can do to help each of those people. A fresh pair of eyes, support, encouragement, or even some relaxing activity getting your mind off the problem, or exercise, all add up.
Perfectionism. Don't let the perfect be the enemy of the good. Surprisingly often, the 'right' way to do something is not apparent or documented, or it's moot, or context-dependent. You have to strike a balance between trying stuff out, prototyping ("code beats arguments"), breaking, failing and learning quickly, vs researching the answer. Academia tends to bias us to the latter and away from the former. v0.1 code is not usually about optimality (although in the occasional cases where near-optimality can be simply and quickly achieved then we should strive for that). By all means come back and polish/simplify/generalize the code as necessary, later, as priorities allow.

To the phantom downvoter, if you have any constructive comment or criticisms please state them

The issue with 2 is that it invalidates learning for the sake of it.

@WeckarE. but so does any pragmatic set of requirements or deadline, or imperfect existing codebase. As per 8., learning does not consist simply of reading other people's experience and advice. One thing I found useful is to keep a set of notes ("Figure out why they implemented X like this..." "What is the recommended way?") that you can return to as time allows.

@WeckarE.: I mean don't take it to extremes, but be mindful when you are going beyond learning the necessary; you can always come back to it later. In my experience 75+% of textbook content were skippable. And when reading a paper, I would skim the introduction, jump to see if there were any conclusions and were they useful-sounding, then scan the methodology, before deciding whether to invest my time. Sometimes the abstract overpromises and underdelivers, sometimes v.v., sometimes the paper is incoherent or needlessly abstruse.

@WeckarE.: they're two types of learning: learning from other people's writings, vs learning yourself by trial-and-error. (Elon Musk, to take one example, is a proponent of the latter and highly critical of the former. Academia tends to predispose us towards the former and disparage the latter. There is a balance.)

A key point is to realise that you can never understand everything and that, to understand a particular topic, you don't need to fully understand its prerequisites.
Your approach is to say that you want to learn A, but A uses aspects of B, so you will learn B first. And then you find that B uses aspects of C, so you will learn C first. Oh, dear, but C uses aspects of D which uses aspect of E which...
You don't need to fully understand B to understand A. You probably just need the basics of B and maybe a couple of sub-areas in more detail. To understand those, you probably need hardly any of C and so little of D that you can, for now, just consider the couple of facts you need there as being knowledge given to you by the gods, which you might never understand.
Notice that we've turned the divergent series A + B + C + D + ... into the convergent series A + B/10 + C/100 + D/1000 + ... .
Later on, if you want to do actual research in A, you'll probably find you need to know a bit more about B, which might mean you need a tiny bit more about C, and maybe you'll have to ask the gods to reveal to you a couple more of the mysteries of D. It's very unlikely that you'll ever need to know that E even exists.

It's rather simple:

Take an introductory textbook on X.
Read it (and do the exercises).
Hard part: Don't switch to another textbook/online guide/etc on any underlying or related subject until you're done with the book. It contains enough of what you need.

Now, choosing a good textbook is not always an easy task but that's a subject for another question. Perhaps get a knowledgeable friend to recommend one that covers underlying/related subjects to some extent so that you wont feel you're learning in a "contextual vacuum".

Like Donald Rumsfeld said:

"[T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – there are things we do not know we don't know."

It is important to know what you don't know. Engineers, e.g., work with a lot of formulas and approximations without questioning every single fact behind it. If you want to work productively, don't question all at once.
You don't need to be an expert in number theory if you want to be good at computer graphics. But you might need Linear Algebra basics and good knowledge of OpenGL. So your numbered list could be used as prioritization. If you pursue your PhD, your goal is to be an expert in your field. And the thing in your list above that interests you most might be that field. Not all at once.

Any answer to this question is necessarily going to be somewhat subjective, as it's ultimately about motivating particular behaviour in yourself.
I certainly wrestle from time to time with "reading vs doing" - reading about new things is fun, but ultimately it's not very helpful (and, to be honest, results in fairly superficial understanding) unless you're doing something with that new knowledge.
For me, what helps with that is choosing something specific I'm going to do with that knowledge.  (Let's face it; "Learning about X" isn't a very well-defined task.)  Now, sometimes it takes a little bit of casting around to figure out something interesting to do in that area, but I can generally find something interesting and tractable-seeming without getting too lost in the weeds.  It could be:

Reproducing the work in some paper which seems interesting
If the paper authors already have made code available, playing with it and trying to extend it in some straightforward way, or fix a bug in it
Committing to teaching some people about a package/technique/paper - coworkers, give a talk at a Meetup or a Papers We Love event, or the like
Choose a set of well defined topics (overview of a couple of recent related papers, a technique, a package) and commit to writing some blog posts about them

etc.  Then it becomes much clearer to me how much time I have to spend digging into related material: "will this help me write the code/patch the code/write the tutorial/write the blog post?"
An added benefit of especially the last three items on the list is that you're now contributing something to the wider community, and making useful contributions is a good way to establish yourself in a research area if you do decide to pursue a Ph.D. or even just do further work.  And yes, paper summaries (particularly compare/contrast recent papers), tutorials, bug fix PRs, etc are useful contributions - just look at how many citations review papers get, or how many people follow Adrian Colyer's tireless work on The Morning Paper.

+1 for "doing something with that new knowledge". Note: you might want to register yourself and use a different user name. user_numeric sucks as a user name.

+1 This is the approach that works the best for me. For example, the OP mentions computer graphics; I would be much more likely to acquire a comprehensive working knowledge about that topic if I had a burning need to, say, illustrate a book for my children or make an animation for my parents' anniversary next year, rather than just a general, theoretical desire to learn about it.

My strategy when I want to learn something new:
1) Think about what you want to accomplish (have the end in mind). For you this might be an animation or game, etc.
2) Go as far as you can on your own. At the beginning this might be nothing, and that's OK.
3) Learn only as much as you need to get to the next step, then go back to step 2. Repeat the cycle.
I find that spending too much time on learning background material results in excessive cognitive overhead. It's often best to start simple and approach learning as a "just in time" process (borrowing from the lean manufacturing principle). Perhaps this is slightly oversimplified, but you probably get the idea.

Exactly. Pick a concrete goal (writing a specific program, proving a specific result, etc.), and only learn from the background material what is absolutely needed to reach the goal.

I used to get this problem when I was first learning programming.
I still get it now sometimes because I have a tendency to want to know everything about a given topic all at once.
Over time though, I've found that the best approach is to learn as little as you can possibly get away with in each subject and then iteratively refine your knowledge as you go.
Your current thought process is

'I need to know all of V before I can learn U'

implying you think you need to learn these topics sequentially and that you think you need to learn each topic in depth before moving on.
What I am proposing is to learn parts of the topics simultaneously or in rotation, i.e.

'I will learn a bit of V and then apply that knowledge to U to
  learn how the two things are related'

For example, you want to learn computer graphics so you decide to focus on OpenGL. OpenGL requires knowledge of vectors and matrices.
Vectors consist of many sub-topics (addition, subtraction, dot product, cross product, scaling) as do Matrices (addition, multiplication, determinants).
Instead of learning all of these topics before attempting OpenGL, learn a sub-topic and then apply that knowledge directly to OpenGL.
Learn what a vector is and how to add/subtract them, then define a vector using OpenGL and test out some addition and subtraction.
Move on to the dot product, test in OpenGL, learn about the cross product, test in OpenGL etc.
By connecting the two parts at each stage, you end up learning the implementation alongside the theory in small chunks. By the time you get halfway, instead of knowing all about vectors and nothing about OpenGL, you know a bit about vectors and how to apply that knowledge in OpenGL, meaning you can do more earlier.

I'm facing this problem in computing all the time.
My personal approach is to decide on something really reasonable I want to achive (like printing "hello world" on screen), chose things that help me towards the bigger goal.
Then I pull all available sources to make that work first. Sources are books, tutorials, stackoverflow searches, youtube videos (a great resource!) or asking someone to help me do it (while I document all steps so that Ican redo it on your own). If I need outside help to solve the problem, I will redo it again from scratch once the person who helped is away!
This gives me either a quick success - or a quick reason to delay or give up!
If I manage, I now decide to attempt to either understand what I did or I set the next goal. Ususally getting things to run involves using code I don't fully understand. I don't mind for the time, because I now have a cookbook example to fall back to. Usually, if I re-read book on the topic after getting something to run, I understand why it's working. Having working code is also helpul if I want to talk about this with somebody else who might be able to help to improve it.

As a different take on it, this also describes classic ADHD (ADD) behaviour (not saying that's you, but its how that condition typically affects people as well), so there are also coping mechanisms from that direction too. A good book on managing distraction might have helpful tips. Maybe as simple as a post-it?
Basically, you have to decide if you are doing these because you need them, or because they interest you. Most people don't need linear algebra to use or enjoy computer graphics code. If you want to, or its for pure interest then more power to you, let your focus wander where it likes. But if you want to do a specific topic, and this is a distraction looking back, then draw a line at some point and know whether or not you want to follow "wherever it goes" or to learn and do something at the first level.
If you have 2 topics (graphics and linear algebra) then you might find you don't need to learn one or know "how it relates" to the other.
If you feel you need this to learn a subject, then the advice given by others is also good - pick a specific target task, and do only what's needed to achieve it. This will teach you some aspects of the toolset you need. After a few tasks you'll get quite adept with tools without distraction.

Yes it does happen all the time - it happened to me a PhD, an academic career, and an industry career notwithstanding.
Please realize that the main value a formal academic training provides in our days of online resources is exactly what you are struggling with here: creating a workflow from learning the basics to gain control of the advanced stuff - all the while juggling inter-dependencies - that will lead you to master your desired skill.
However, it is clear that if you wanted to go back to school, you would already have done so.
So you'll have to go the way we all go after college: start from the very end and practice the simple stuff of the skill you would like to master. Realize you don't really know what you are doing, study it a bit and try cracking the nut again. You'll think you got it, while all of a sudden a new problem will have you face the harsh truth there's so much you don't know yet. Back to a book, crack a new problem, hit the wall with another one. Live, die, repeat. Don't obsess with going over all nine hundred pages of computer graphics bible. You'll probably need only two or three chapters anyway.
That's the way it goes in adult age. Luckily with computers-related problems, there's plenty of companies that are willing to pay for you to learn all of this. Unlike academia :)

This is meeeeee, oh, man. I'm such a dilettante. I forgive myself for it if it's hobbyist stuff and it just turns out not to be a real passion. It doesn't always feel good, though, and I get you. When I'm doing that, I also complicate matters by holding myself to the standards of what I think other people can do. For instance, I keep trying to tell myself that, if I were really interested or capable, it should be easy to make myself do it and stick with one thing. That's not true, though, so I've learned to treat it differently. Pick a very specific topic that interests you. Go ahead and do some Googling, but sit down and tell yourself, "I'm going to find a good book to learn about [that very specific thing] and buy it." Don't put it on your Amazon wish list. Go ahead and order it. Then, when it comes, make yourself a rough plan for how you want to go through the book. Come up with at least an idea of how often you want to work on it, and be conservative so it's easy to maintain. Then, dig out a bag, get yourself a clean notepad and pen or pencil, and put those and your book in the bag. That's now your school bag. Nothing else goes in it except your wallet, phone, and keys. At the times you've set out, take your bag and go to a coffee shop or library for a limited period of time (plan on an hour the first time and see how you feel, but decide the period ahead of time and stick to it).
People like us have a tendency to burn ourselves out, both in the planning and in the execution, so if you work on avoiding burnout, you'll go a long way towards being able to accomplish your goals. I'll add that I've also had a lot of success with MOOCs, because they keep me on track.


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My master thesis supervisor refused to recommend me for a PhD
He reasoned that I lacked the understanding of the subject and so not competent enough for a PhD. I agree to his concerns, since I was studying this concept for the first time (for my thesis) and was a bit slow in my approach. On the other hand I'm very passionate about this particular subject and intend to pursue an academic career in this field. I did manage to get a decent score in my thesis and graduated with cum laude. The question is should I give up and take up some job since nothing will move forward without a recommendation.

Try other advisors.

I'm very passionate about this particular subject — Passion is not enough.  You actually have to be good at the subject if you want any hope of pursuing an academic career.

With apologies to The Incredibles, If everyone gets a recommendation, no one does (because the recommendation is meaningless)

You could consider doing an internship before a PhD, to get a new chance of earning a recommendation letter.

Agreed with @JeffE.  I want to be careful: we don't have nearly enough information to give you meaningfully discouraging advice.  However your advisor does, and what you say to counterbalance his assessment is not of equal weight.  Having a passion for academia is a necessary condition for an academic career but it is not enough: think how many people are passionate about, say, baseball and how few people become professional baseball players.  It is not rational to say "I must have an academic career, so I'll work towards it one way or another."  You don't need one and need not get one.

You should still apply. I went to a for profit and was still accepted to a prestigious university because of my GRE score and strong application. (I did get many rejections for not attending a 4 year institution) but others made the exception. Plus you'd regret that you never attempted to make yourself an expert in the field you care about.

How can you graduate "cum laude" when your advisor obviously doesn't support that designation?

From what you've written, it sounds like your adviser doesn't think you know enough to be able to research a subject. Knowledge can always be learned. It's actually a very important trait for a PhD student to be able to sit down and learn what they need on their own.
There's no reason not to apply to PhD programs. It's going to be very hard to get into one without a letter from your adviser, but it's not theoretically impossible. The worst thing that can happen is that you spent some time and money and fail to get accepted anywhere. The best thing that can happen is that you get into a program and have the chance to follow your passion.
On advice from Pete L. Clark, I'm adding a caveat: You should be careful as to what sort of programs you're applying to and think carefully about whether those programs will lead to your desired outcome. Going into a PhD program is not something to be taken lightly. It will takes years of your life and be a potential drain on you mentally, emotionally, and economically. To land a decent academic position you need to graduate from  a top program and produce high level research. Afterwards you may have to put in more time and effort as a post-doc. Once you do finally get a position, you'll have to work your way through the tenure process.

"The worst thing that can happen is that you spent some time and money and fail to get accepted anywhere." I'm afraid I disagree.  I think it would be worse for the OP to be admitted to a poor PhD program, having to pay funding out of his own pocket, and spend up to ten more years realizing that he cannot complete the program and/or cannot have an academic career afterwards.  Anyone with a master's degree in something has already followed their passion in that thing and should have enough training to continue to do so on their own.  Almost no one wants a PhD and not a career that uses it.

I disagree. There are jobs out there that require a PhD. I'm assuming that OP wants one of those. I'm also assuming they're smart enough to drop out of the program if it goes poorly. The only indication that OP is not a good fit for a PhD is that one professor said they lacked some understanding of a subject. Without knowing the full details, it's hard to read much into that statement.

"There are jobs out there that require a PhD. I'm assuming that OP wants one of those." There is no need to assume: the OP says he wants an academic career.  It is an empirical fact that only the upper echelon of PhD students can have a successful academic career. "I'm also assuming they're smart enough to drop out of the program if it goes poorly." After how many years and how much of their own money do you assume it will take for someone to drop out?  And what will the indication be, if it is more serious than their thesis advisor recommending that they not continue?

I agree (and said in a previous comment) that we cannot know whether the OP is suitable for an academic career.  But you seem to advise everyone who is interested in an academic career to attend whatever PhD program will admit them.  I think that is not just bad advice but rather irresponsible.  Certainly anyone contemplating a PhD program and an academic career should take steps to evaluate the chances of success.  The OP should take more steps in his evaluation, not fewer.

@PeteL.Clark I've added a larger caveat to the end of my answer based on our discussion. Thank you for bringing those issues up.

I appreciate your revisions, and I have changed my downvote to an upvote.

While I was an undergraduate student at a lower ranked institution in my country, all the professors in the department were thinking that I don't have any academic merit. Just after that I took some time outside the school, and later attended classes at a top-ranked school in my country just as a visiting student.
I was the best in the class and the professor was overly blunt to state that fact in front of the other students. The people in my undergraduate institution were so biased and mediocre that they have never had an intention to give me a chance or were so insidious and sinister to undermine me.
So, sometimes, at some places you simply do not fit with others. You should try your chance at other places, internships etc. A single institution or professor does not suffice to assess your abilities.
A last note: some academics are real jerks, do not give up at the first obstacle.

I would suggested that you edit out the offensive part in the last sentence.

Letter of recommendation is not always mandatory for all PhD admissions. You could still apply for a PhD without it.
You might have to vouch for your talents in other ways to certify your ability. Eg., Online certification, skills development, and independent research.

Saying that you "lack understanding" seems like a really vague criticism, especially when you are working on a specific topic for the first time.  Ask other professors what they think.

There are two aspects of this.  One is Should I forget about a PhD? and another is Is this a show stopper?
But for both questions, a lot depends on the opinions of other elders in your department.  Are there other professors in your department who know your work, whom you could approach for career advice?
When you meet with such a professor, try to keep your questions open ended.  It's okay to express an interest in pursuing further studies, but in that preliminary conversation try to keep all options on the table, even if in your heart you feel a strong drive to jump straight into further studies.
Unless, of course, the professor asks you what your preference is.

The question is should I give up and take up some job since nothing will move forward without a recommendation.

You are right, a recommendation from your thesis adviser plays an important part in getting a PhD position (experience!). But giving up is out of question if your words I'm very passionate comes from your heart. I strongly disagree with those who commented, one have to be good at the subject if you want any hope of pursing an academic career. It bear a high degree of vagueness with it because the terms good and hope are incredibly inappropriate in this context as it is not at all clear that OP indeed lacks the potential to be good at the subject.
I think you can do a couple of things here

Talk to your thesis adviser and ask him politely to explain why he thinks you are not apt for the recommendation.
Value your adviser's words and spent some time on working out yourself on the subject core.
If you think that your adviser misjudged you, try for a short term project position, which can be relatively easy to find even without your adviser's recommendation.

I think there are plenty of choices before you and tremendous room for improvement, once again in my humble opinion, its too early to give up.

@Down voters, at least show the courtesy to let the answered know why you down voted.

Is it out of the question for anyone who is passionate about space exploration to give up the idea of becoming an astronaut?  For anyone who is passionate about athletics to become an olympic athlete?  Disagreeing with the statement that you have to be good at the subject to succeed in academia is disagreeing with plain common sense.  Telling someone that you don't know that it is too early to give up is what is inappropriate.  If for instance all the faculty members the OP has ever interacted with think that he should not go to a PhD program, then he should give careful thought to giving up.

@PeteL.Clark Its clear that you haven't read the answer fully. Of course one have to be good at the subject to succeed in academia. One may not be good right now. But that doesn't mean one won't be good, on the contrary he will be good if he have the potential and the passion to work hard for the subject.

"Its clear that you haven't read the answer fully." I paid you the courtesy of explaining my downvote, and you replied with a bad faith assumption about me.  If you want people to be courteous to you, please be courteous in return.  For now I will not engage with the rest of your comments.

Downvoted. I never said OP should give up.  Neither one of us knows whether OP is good—or has the potential to become good—at their subject or not.  Nevertheless I stand by my comment.  If you aren't demonstrably good at your chosen field of study, you will not get an academic job.  Suggesting otherwise is irresponsible.

I agree with @Sathyam Recommendations are important, but I would add that OP should ask from his/her advisor to write explanation statement, ( something like progress report) where all the points that are missing in OP knowledge and preparedness for PhD and overall Master progress will be in great detail explained, I dont know if in USA have this, but in my uni it is usual that every advisor write a report about positive and negative sides of their advisees

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A PhD program recommendation fell through. How do I deal with this?
I am finishing up a MA in Global Affairs, and I am in the process of applying to several doctoral programs. Two and a half weeks ago, I asked three of my professors if they would recommend me. I made sure to note in my emails to them that if they chose to support me, one of the program deadlines was substantially sooner than the others. Two responded positively almost immediately, but one did not. However, he has encouraged me on numerous occasions to apply, and speaks very highly of me. Needless to say, I was very surprised when he responded 5 days before the first deadline to say that he was not prepared to recommend me. He said that he normally only write recommendations for students who have served as his GA, and that if I want him to write one for me I have to draft the letter I expect him to write. I said I could do that... but would we be able to do this in time for the upcoming deadline? He said no, and there was no way I could ask someone else with such short notice.
However, I really do not know what I should say to the other people who recommended me. I know they have already submitted their letters. Do I say that I simply couldn't get my act together in time, and risk looking unprepared? Do I tell them that one of my recommendations fell through, and look like I must have done something to change that person's mind? Do I tell the full story, and look like I am just trying to place blame elsewhere? Or, do I just not tell them at all? They all know each other, so they could find out even if I don't say anything. None of the options seems great, and I am kind of at a loss.
Edit to add (taken from comment): My main concern is that asking someone else to submit something last minute or late might not be appropriate.

Get another letter. Grad app deadlines are many times not as strict for letters as they are for your application materials.

and there was no way I could ask someone else with such short notice.

This may be the conclusion of your analysis but in my opinion it doesn't hurt if you'd actually try to contact another potential referee. I agree with Prof. Jose. F. Martinez in his answer and I would like to add that this should not be your only reason for not applying. I think your questions reflects concerns in the not so near future that you can actually try to find another person for the recommendation.

Actually, I am not at all concerned about never being able to find another person to recommend me. I have another academic reference who has agreed to write letters for applications that are at a later deadline. My concern was that asking them to submit something last minute or late would not be appropriate.

@lauren31428 Well, I must say this is not at all clear from the OP. there was no way I could ask someone else with such short notice, I either misinterpreted it or the subsequent questions followed couldn't communicate the above mentioned concern properly. Either way, in my opinion I don't think there is anything inappropriate in asking a new potential referee on a relatively short notice such as this given the premise that it is not at all your fault. I would explain why if I were you.

As I said, it is likely that that third reference, if submitted a bit late to the institution in question, will be taken. (In any case you lose nothing by trying.)

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Do both first-author and co-author publications from a potential advisor imply current research activity?
I want to apply as a PhD candidate and therefore looking up on various potential supervisors who have common research interests. My approach is by first visiting their webpages, looking up on the Research Interests mentioned and then checking their recent publications on a particular topic I am interested in. I find many, but in most of them the potential supervisor is a second or a co-author.
In order to gauge the current active research field and work of the supervisor, must they  be the first author or does being a co-author imply the same?
If this matters, my subject is Engineering and Technology.

My PhD advisor was never the first author on any paper with me.

If they were the first author too often, that would suggest you wouldn't get to be first author if you worked with them.

An advisor that steps back and allows his students to be first author is giving the limelight to that student and shows that he is more interested in your success than his career.  This is an indication, but not a guarantee, of a good advisor.

Related: https://academia.stackexchange.com/a/2474/19607

What is your field? The order of authors has a different meaning in different fields. For example, in biology, a supervisor is never first author and instead has the very prestigious last position. One is expected to move from being first author (the person who did the work, usually a student or young researcher) to being the last author (the person who supervised the work and lead the team). In math and related fields, the authors are just listed alphabetically. So please tell us your field.

This seems ok since their students spend more time, perform the ideas, and do implementations. So their name comes before the name of their supervisors. Additionally, at some departments it is common write the names lexicographically.

@terdon I mentioned my field in my question; it is Engineering ,to be specific Electrical engineering

Of for... Yes you did, and it was right there, from the beginning, in its own paragraph! Sorry, I managed to miss it completely.

This is very common. The professor has a research group comprised of students & postdocs (and possibly collaborators from other institutions), who are all working on different problems that the professor finds interesting. When those projects are done, a paper is written with the student/postdoc as the first author and the professor as a co-author. This is normal, since the student/postdoc does most of the work, but the professor is involved anyway as the supervisor.
Since the professor is involved in all these papers, you can gauge their research field by looking at the papers, even though they aren't the first author.

It is surely field dependent, but I do think that the custom of advisor/research-group-leader as last author (and also corresponding author) is not uncommon and pretty good system for a number of fields.

Well my supervisor is also mostly second author in her publications, since, she works in a team where her principal investigator is the first and main author. This does not mean my supervisors contribution is any less.
So, there are few things check, you could check.

In the publications, where your potential supervisor is second author, is the first author always same?

Check if he/she is involved in a big project and who is leading that project.

How can you correlate the possibility of the order in the author list and the research contribution?

Try to gauge the research out of the publications, for instance, international collaborations, workshops, talks etc

I would say that if they mention it in their research interests (which you've already checked) AND they consistently publish papers in the field (even if not as primary author) then that is a research interest.
It also means that, if you apply and get in and work with the professor on papers, you may get to be first author, which is nice for many things. You certainly want a mentor who is interested in what you are interested in, but you also want one who is nice, and this is one indication of that.

There may be a few reasons why advisors are second authors. 1) Faculty has their own research group and they are supervising students from their research group. In this case, supervised students are primary author and faculty are second author. 2) Faculty is collaborating with other people who are more actively working on paper rather than the faculty and my first point applies here too. 3) It depends upon norms in the field—some field list advisor as last author and other as second author. So it’s not easy to generalize.
Coming to your most recent works question—since faculty are already publishing, their research might be ongoing and they may already have team set up to work on the project and highly unlikely to hire new students because students are bounded by two obligations of research and education. So, it’s highly unlikely student would drop or leave research team before finishing research.
Best way to approach faculty is to email them with CV attached if your broader field/interest matches with the interest of faculty. If faculty is willing to hire students, they will respond. Sometime faculty are willing to apply grants if your idea is good enough. So, best way is to email them. Also, visit society website if there is any—some jobs are posted there.

In some fields (like my bit of Physics, overlapping with engineering) , the supervisor is usually the last author. That's last in order but not necessarily least in contribution, and a marker of their role.
So my PhD first author papers have me, other authors, my supervisor, where other authors includes the relevant postdocs, other PhDs students who contributed, and external collaborators.  Similarly I've been in that "other" category as a result of other contributions, both during my PhD and as a postdoc.
This, along with similar authorship patterns, is a sign of a group with a healthy culture of working together (not a guarantee, just a sign).  It indicates that credit is being distributed, hopefully to those who most deserve it.
One thing to watch out for is supervisors who have done enough work - normally experimental design + writing) to be first author, but allow that position to go to the student, who needs it more.  That's only mildly generous, because supervising successful students is important to a PI's standing and it's a way of getting the publication that the PI needs, but it's not necessarily good for their students' training.

Really? I thought Physics and Engineering took the Math approach and listed authors alphabetically. Is this ordering specific to your particular sub field or general to Physics and/or Engineering?

@terdon I can't speak for everyone else, but in the semiconductors/spectroscopy areas I'm used to, author order isn't alphabetic.  When I've contributed in adjacent fields, it's been the same as I'm used to as well.  AIP and IOP guidance implies the authors have to sort out a sequence,  IEEE "leaves the order of authors to the discretion of the authors"

@terdon In physics there are sub-field-dependent practices, see this.

Here is the short answer to your question:
In almost every field where author order matters (i.e., is non-alphabetical), the first author is typically a junior researcher and the last author a senior researcher.  The first author is the one who "did most of the work", while the last author is the one who came up with the ideas and directed the project.
Therefore, when scouting potential advisors, you should look for their last author publications, not their first author publications.

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Applying to PhD programs with only 3 years of undergrad in the US
I am an undergraduate at UC Berkeley majoring in statistics. I can graduate in three years because of high school credits. I originally thought I should add another major or just take lots of extra classes in order to be in undergrad for four years. I did some statistical research relating to certain social sciences, and this really inspired me to want to continue with research, so I am now hoping to apply to statistics PhD programs once I finish undergraduate.
Is it better or worse for me to graduate in three years or four years in terms of PhD applications? It seems that I can just finish my PhD research a year earlier and spend a whole years' less on tuition. On the flip side, I may not have as much experience when applying, and although I would have completed my major, there might be other useful courses that I will potentially miss out on.
The last option -- is this viable? -- is to apply in my 3rd year, and if I don't get into any programs, apply again after a fourth year.
Might graduating in three years hurt my PhD admission chances?

If the courses you did take were the standard set for your major, and you did well in them, what is the worry?

Also related: Better chances for math grad school: graduate in three years or four?

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Authorship for Publishing Master’s Research Paper – First Author vs. Second Author?
My tutor has approached me with an opportunity to publish my master’s dissertation research, which examines the relationship between social media sentiment and cryptocurrency dynamics. I’ve been given two options regarding how the paper will be written and how authorship will be credited:

I write the paper myself using my portfolio and final report as a foundation. My tutor would be the corresponding author and second author, while I’d be named first author. This option would involve writing a formal research paper (including a literature review, methodology, results, etc.) and also making some technical improvements to the analysis.
My tutor writes the paper, making all necessary improvements and finalizing the submission. I’d be listed as second author, and my tutor would be first author and corresponding author.

I don’t plan on pursuing a career in academia or publishing a lot of research papers in the future, but I do see value in having at least one paper published to enhance my CV. I understand that being first author holds more prestige, but the time and stress involved in doing all the writing and technical adjustments myself might be a bit overwhelming.
Is first authorship worth the extra workload if I’m not looking to pursue academia long-term? How important would second authorship be in terms of the recognition it offers, especially for non-academic careers?

Given your relative lack of experience in writing academic papers and your lack of interest in a career in academia, I'd suggest that 2 should be fine. It would be better if there is a "contributions" section in which it is clear that the work was originally yours.
Corresponding author is a job, not an honor, and the tutor is probably better prepared for that.
I would suggest, however, staying connected in the process, reading all drafts and making suggestions as needed. That will give you some experience being adjacent to the process while not responsible for it.
I would answer differently, however, if you plan a career in academia. Lot of work, yes, a bit more prestige, yes, but the experience is the more important thing.
Your tutor would probably find first authorship more valuable than you would (given what you say) and a certain amount of generosity in such things is valuable since it opens the door to future collaboration should you change directions.

As @Buffy already mentioned, if the space allows, it is better to add a contribution section such as: (Author1: draft writing, conceptualization, implementation, evaluation, ...; Author2: supervision, paper writing, ...). With this, you preserve your contribution forever; and the position does not really matter (unless formally required, like in PhD regulation). However, if you plan to submit to a conference, you might not have enough space to add this section.


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How are trustees chosen?
I thought that trustees of a university are leading people in the region or around the country (politicians, industrial leaders, donators, etc). I accidentally saw the Board of Trustees of Princeton University, and was surprised that one of the members is just a student from California.
It makes me wonder, how the trustees are selected? and what are the requirements for becoming a trustee?

So, exactly how does one 'accidentally' see the Board of Trustees?  Just wondering...

Two cases that I happen to know about, as an employee or alumnus.  The  University of Michigan has a Board of Regents elected by the voters of the state of Michigan (usually very far down on the ballot, where not too many people bother to vote).  Harvard University has a Board of Overseers elected by Harvard's alumni.

To be clear, the trustee in question is not "just a student from California"; she graduated from Princeton 4 years ago and subsequently enrolled in a Ph.D. program at Stanford. When you say it this way, it's much less surprising...

Varies by institution.

Every university should have a documented process for choosing trustees, regents, etc. Ours are appointed by the Governor. See the link for what Princeton does. There are students on the Princeton board because a) there was a grad student among the 13 elected alumni trustees (2011 grad), and b) because there is always a graduating senior elected. Both of these kinds of alumni trustees serve for 4-year terms.


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What do search committees really read in a CV during initial screening for a faculty position?
Faculty positions are very competitive, and each advertisement should roughly receive hundreds of applications.
I think search committees cannot spend more than 1 min reading each CV during the initial screening (correct me if I'm wrong).
CVs are usually 10-20 pages documented in different formats, and it is not easy to capture potentials of a person in 1 min.
Owing to the fact that not all committee members are fully familiar with the journals in which applicant has published, his/her research impact, his/her universities of education, or the importance of his unusual achievements.
Then, how does a search committee shortlist the candidates during the initial screening? What do they quickly look for in a CV to keep the candidate?
NOTE: My assumption for 1 min for each CV was based on a simple math. If a job ad receives 400 applications, each member should spend almost 7 hours to review only CVs (not looking at other documents). If my assumption was somehow wrong, please forgive me. Your answer can clarify the issue anyway.

"What do search committees really read in a CV during initial screening for a faculty position?" In computer science in Europe: the link to Google Scholar. All the rest of the text will be looked at, if at all, at a later point in the process.

I'm serving on a hiring committee this year that received roughly 180 applications for a tenure track position in mathematics.  All of the members of the committee reviewed all of the applications.  Most could be very easily eliminated from consideration (PhD not completed yet, PhD in a field other than mathematics, wrong area, etc.)  Although I got through some of these easy applications in a minute, the better applications took considerably more time to review, and I ended up spending about 15 hours reviewing applications for the first round or an average of about 5 minutes per application.
When the committee met, we wrote down our top 20 lists, and it was relatively easy to select phone interview candidates, since only 18 candidates were on the top 20 lists of at least two of the committee members.  We offered phone interviews to 18 candidates and had 14 candidates who agreed to phone interview.
I don't blame applicants for submitting to lots of places, but the reality in mathematics today is that nearly all institutions will be able to find enough applicants to phone interview that meet all of their requirements, are in the preferred area of specialization, and have some post PhD experience (post-doc, VAP, or both.)    If you don't have these qualities, then you're not likely to get through the first round of screening.

I am both surprised and impressed that every member of your hiring committee read all 180 applications.  However, at a certain point this becomes prohibitively impractical: if next year you got 360 applications, would you still read them all?

From what I have heard, 180 applications for a tenure-track math position in the US seems very low, at least for departments that used MathJobs.

@NateEldredge our advertisement was fairly specific and there are various things that make the position not highly desirable (a 3-2 teaching load, small town location, at a public institution where salaries aren't great.)  I think that if we had many more applications we would certainly have divided  the pool up and had only 2 or 3 of the committee members look at each application, with some kind of randomized sampling scheme.

I'm most surprised by the requirement that the PhD be only in mathematics. Do math departments not give consideration to non-math PhD's, or was this because of the nature of the specific job?

@aeismail: All the tenure-track faculty postings in my (math, as you know) department specify that the candidate should have a PhD "in mathematics or a closely related discipline".  Once I asked a senior colleague what would count as "closely related" and his answer was essentially that we would cross that bridge when we came to it.  I would estimate that well over 90% of our applicants have (or say they will have in time to take the job) a PhD in mathematics.

@PeteL.Clark: Would statistics count?

@Mehrdad: Statistics is part of some math departments (though not mine), so: it certainly would for me, yes.

The position in question is in applied mathematics, so a PhD in mathematics, applied mathematics, applied and computational mathematics, etc. would clearly be appropriate.  Our department has PhD's in statistics, but that wouldn't be appropriate for this particular position.  PhD's in Physics, Chemistry, etc. are also out of bounds.

Excuse my ignorance, but what is a VAP?

VAP="visiting assistant professor"  It has become common for new PhD's in mathematics to work in short term non tenure track faculty positions with full teaching loads for a few years before moving into a tenure track position. This is different from a research post-doc where the job is primarily research (although their may be some teaching in addition to research in many math post-doc positions.)

Let me make a few points with regards to to your solution of this Fermi problem.
Hiring committees have more than one member.  The more applications received, the more members the committee usually has.  Moreover, it is highly unlikely that every committee member looks at every application in the first round.  Exactly what is done must depend strongly on the department, but I can describe my own experience.  In my department -- mathematics, UGA -- we do usually get 300-400 applications for a tenure-track job opening (sometimes for two or three positions at once, but that doesn't help any!).  The applications are processed by subject area.  In fact in my department the "hiring committee" consists of all tenure-track faculty, so initially I look at applications in my field -- number theory -- and closely related fields.  This usually gives me about 50 applications to look at.  I look at them as they come in, not all at once at the end, so in the first phase I probably spend something like half an hour a day over the course of several weeks looking at applications.  This gives me more like 10 minutes per application, on average.
Another point is that CVs can be 10-20 pages in length, but this is usually for people towards the end of an academic career who insist on putting every talk they ever gave and class they ever taught on their CV.  For applicants for an assistant professor position, I would expect CVs to be in the 2-6 page range.  (I am a 2003 PhD.  My CV is holding steady at 8 pages.)  Anyway, it is important to organize your CV carefully and put the information in the places that people are expecting to find it.
Finally, I want to say that amount of time spent on each application is far from uniform.  I said above that I spend about 10 minutes per application on each of 50 applications.  But most applications I am spending either more than 10 minutes on -- it takes more like half an hour to read an entire application, no matter how quickly -- or much less.  I do try to read every application in my field however briefly, but "however briefly" can be awfully brief.  Hundreds of applications can, and must, be eliminated essentially immediately.
Added: I see that I didn't fully answer the question asked.  In a quick first pass on applications, I look at more than just the CV.  Our applications come to us in a very convenient online format via the MathJobs service, so that one click brings to a screen where we see the candidate's basic information (name, date of PhD, PhD institution, PhD advisor, current institution) and can click on various documents they (and/or others) have uploaded, which generally include:

cover letter
CV
research statement
teaching statement
publication list (sometimes)
multiple recommendation letters

I usually look first at when, where and from whom they got their PhD, and then I click on the CV to see their background and their publication list.  If I really don't like what I see -- e.g. no PhD, PhD is not in mathematics or something very closely related, clearly insufficient publications -- then I stop right there.  This is maybe 20% of the applications.  The next thing I do is to quickly read the advisor's letter and the research statement.  Based on that information I decide whether to read on.  (Maybe I've eliminated 50% of the applications at this point.)  I next read the other letters and start to look more carefully at the candidate's research.

Would the requirement of a PhD in math hold even for applied math positions? And, if so, why is there such a bias?

@aeismail: The language has been the same in all full-time position announcements I've seen.  We have some applied math hires who are half-time in other departments.  Their PhDs were in math...but if they were in the other discipline it would probably count as a closely related field.  I'm not sure what you mean by bias: what plausible candidates do you imagine are being excluded?

@aeismail: A few years ago, our applied math group proposed someone without a Ph.D. in math for a faculty position, and people on the pure side objected that the candidate's Ph.D. wasn't in math. Whether this was fair or not I can say, but it indeed seems fairly atypical for math professors to have Ph.D.'s in anything other than math. (Or in "applied math"; in some universities this is a separate department.)

If faced with 200+ applicants, the committee would probably perform an "initial cull" where each of the about 5 members would select maybe 10 candidates to look at more closely. That would eliminate numerous applicants who clearly didn't fit the description (wrong specialization, no publications if the position is research-intensive, no teaching experience if that really matters for the post, not-appropriate alma mater i.e. that place is a degree-mill or on the wrong side of a political divide). In anticipation of a large applicant pool, I've known departments to request a 1-page essentials-vita as well as the full CV (highest degree, major employment, publications, conferences and grants), which makes it easy to home in on the half-dozen who might make the cut. The committee will probably miss out on some great candidates with significant latent potential.

I have been on faculty search committees at an R1 math department, where our search was targeted to a research area somewhat close to mine. It generally takes a minute to find out the answers to the following questions:

Has the candidate produced a substantial number of papers, and published at least some of them in high-quality journals? Alternatively, has the candidate published at least a couple of papers in extremely good journals? (Our candidates are required to submit a publication list, so finding out this information is immediate.)
Is the candidate doing research in a subject area that I am particularly interested in?
Are any of the letter writers people I know and trust, such that I would be especially inclined to interview someone whom they recommended?
Skimming the thesis advisor's letter and one or two more, does the writer have anything credible and strongly positive to say about the applicant's work?

If the answer to any of these questions is yes, I give the application a more careful reading.

The last time I did this the piles were divided so that every application was seen by two members of the committee during the first round (described as a "well, duh!" sort), and needed an up-check from both to survive. We were looking for easy reasons to reject an application.
I read the cover-letter and scanned the CV and looked to see that the other documents were present.
Things I used to reject packets

Documents given generic filenames (i.e. "CV", "TeachingStatement") instead of containing the applicant's name (thereby forcing me to rename them) was something I could overlook but it was a useful channel marker.
Generic cover-letter (or on two occasions a letter intended for a different institution).
Lack of suitable experience (we're an all-undergrad teaching-heavy department, so that means ditch application with neither teaching experience nor signs of interest in teaching). I was also interested to see that the candidate has some publications, but it was just an existence check.
Multiple and blatant errors in spelling and basic grammar. I was prepared to be tolerant on this front for applications from foreign parts, but there was no need.
Missing parts of the packet with no reasonable explanation given in the cover letter.

The surprise to me was how many this culled. These are the same things that every placement councilor in the known world harps on about. I'd have thought people applying to be professors would have that bit down cold.
In the second round everyone looked at them and we actually sorted into "great", "good", "maybe" and "no".
Then we started calling references for the two highest ranked piles, before a final cull down to a short list.

The file naming item only seems reasonable if the applicant was given file naming instructions.

I respectfully disagree: when you put the packet in the email you know it is going to end up on someone's computer with a lot of similar files. This is a basic courtesy and attention to detail similar to being on time for a in-person meeting. And as I said, it correlated strongly with other basic goofs. None the less, I didn't bin an application just for that.

It's an implementation detail of your system. If you're receiving hundreds of job applications by email, might be time to agitate for your institution to invest in access to an online job application portal that manages applicant files for you--rendering file naming largely irrelevant. I agree with an email application, naming a document with generic names might show some lack of perspective taking, but it could be your colleagues create a folder for each applicant and spend their time renaming non-generic files to have the generic names. Either is arguably a logical system.

I wondering if you have any advices on how to not write a generic cover letter. Some schools/departments, including many excellent ones, don't have anything distinctive that I can easily talk about. I have looked at many departments, and for many of them the best I can think of is that they have some faculty members whom I am interested to collaborate with.

@CCcat I looked for generic language for the department names (ours is slightly unusual), unspecific description of the position on offer and a lack of any acknowledgement of which school they were addressing. That's a pretty low bar.

@dmckee I agree. This is a pretty low bar. Anyone should get those right.

In economics, for assistant prof position, they read which university the PhD is from, the quality and number of publications or number of working papers. This is from the CV at a glance, 20 seconds.
In addition, the main advisor's recommendation letter contains a sentence like "I recommend this candidate to all universities outside the top n" or "I recommend this candidate to all universities without exception". The larger n, the worse the candidate. Finding this sentence in the reference letter takes less than 30 seconds on average.
Overall, the initial evaluation fits in one minute.


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Is it acceptable to ask how a professor would like to be addressed through email?
I am an undergraduate student working for a professor this summer. Up to this point, I have referred to the professor as Dr. LastName. I have met his research group; his graduate and PhD students refer to him by his first name. Moreover, he signs his emails to me with his first name.
Is it acceptable to ask, through email, how he would like to be addressed? Or is it proper etiquette to wait for him to correct me? I would prefer to ask him in person, but I will not be seeing him for several weeks.
For example, in an email: "I have noticed that you sign your emails as 'FirstName'. Would you prefer that I address you as 'FirstName'?"

In my experience, you should always refer to a professor by Dr. Lastname as an undergrad. The professor might sign with their first name only but it doesn't necessarily mean they want you to address them by first name. Addressing a professor by first name suggests that you're their equal. Younger professors will sometimes prefer that their grad students call them by first name since they feel a bit uneasy about the level of formality and feel that their grad students are not really their junior in a meaningful sense. It doesn't hurt to ask, but I'd be surprised if they were okay with it.

@CameronWilliams you should have just written that as an answer.

My approach to this 'problem' on email (as a grad student not an undergrad) is to start with "Dear ProperTitle LastName" on the first email and to follow that up with "Dear FirstName" if the response to the first email is signed off with FirstName and to continue with "Dear ProperTitle LastName" if they sign their name in that way (which incidentally has never happened). Whether or not undergrads should be more formal I'm not sure.

@CameronWilliams Anyone who signs their emails FirstName, and expects not to be called that is being totally unreasonable.  If I'm emailing with someone I'm not yet ready to go to first names with, I sign with my full name or "-BW".  (I' ve never signed an email "Dr. Webster" or "Prof. Webster", and don't intend to start now).

This seems very depending on the culture of university and field. During my undergraduate forays in the USA, in Romance Languages it was always Dr. L'astnamé and in all my time in math it was always FirstName. Here in UK it's almost universally Dr. L'astnamé.

Addressing a professor by first name suggests that you're their equal. -- Yes. And this is precisely why I want students to address me by my first name.

In contrast to other answers voiced here, to me, signing a mail with {Firstname} is a clear indication that one can address the sender by {Firstname}. It would be unreasonable for the prof signing with {Firstname} to expect otherwise. I don't agree with the "always better to play safe"-advice, as I strongly believe in that an academic discussion should be based on arguments, not on status. If I expected my students to call me in a formal way, then I would have to address them in a more formal way as well (at least, that's what I feel). When I indicate that I as prof feel comfortable with the informal route (which I do by signing my mails in this way), there is no reason not to accept that offer.
Having said that, it is of course highly dependent on culture and language.

Yes,
it is completely acceptable to ask how you should refer to someone via email.
In fact, I think I still have the email I sent to my supervisor.
Here is the (edited) last line of one of my early emails while sorting out our first meeting:

I should also ask, what is your prefered form of address? Would you
  prefer I address you as Dr. L'astnamé, or by first name, or in some other
  way?

The response was:

Regarding forms of address I think Dr. L'astnamé will be fine

and so that is what I called him all through my undergrad project, as a short-term research assistant under him, and now into my Ph.D.
I'm sure he will tell me if he would like me to change.
It is generally acceptable to ask people matters of etiquette concerning them.
(Life would be hard if asking what was polite was normally impolite.)

It is almost never impolite to ask how to be polite.

but it can be kind of a hassle :P Many professors will find it tedious to deal with matters of etiquette.

Since each student will, at most, ask this once and it's easily answered, I think most professors will, at worst, sigh and reply.

If you're really worried, find the department administrative office and as them how Professor Whatsis likes to be addressed. They probably know

Interesting question... Let's put aside opinions and try to list the facts:
1. Variability
The preferred for of addressing a prof, much like anyone else, will vary immensely. It ultimately depends on many factors like:

culture in that country ...
and in the country of origin, if the prof in question has a different background
culture in that particular faculty/department (usually less formality/honourifics in Maths/CS/Physics etc)
how down-to-earth that particular person is
...

2. Uncertainty
If you don't know something regarding another person, you essentially need to acquire that piece of information. It won't magically dawn on you... So your options are either to ask that person, or to ask someone else that knows this person.
The risk is, if you ask another person, you might be making the assumption that the person you ask has a different level of relationship with the prof. It might be so that the grad student you ask "goes way back" with the prof, and while they might be on a first name basis, you might not have the same grounds to stand on.

Bottomline: If you don't ask you can't know the right way to address the person. Noone can really blame you for asking how to refer to a person, especially if they are at a higher position than you, implicitly or explicitly.
You might be considered unnecessarily formal, or stiff, but still it's better than making an assumption and making a rude mistake, especially if the person in question comes from a culture where titles are taken very seriously.