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Not including student who contributed very little as coauthor of paper
I am a researcher in condensed-matter physics, and I recently started a research project with another professor, for which I hired a student for a six-month undergraduate internship. The student was under my supervision. During the internship she showed good motivation, but did not make any actual contribution to the progress of the project.
Finally, she started a Ph.D. with someone else. I personally took over the project, taking care of both technical and conceptual parts, which requires a lot of time and work.
I kindly invited her multiple times to double check some of my work, but she had not done anything on the project since the end of the internship.
In the past few months she started having an attitude that I find disturbing. She replies to emails once in a while, proposing new conceptual points to explore, as if I were the student and she was the supervisor. On top of that, she ignores the invitations to double check my calculations, which would require actual work on her side.
Given that she did not make any actual contribution to the project, I am inclined to not include her as an author of any eventual publication. I think that it would be definitely fair to include her in the acknowledgments, but unfair to include her as an author.
What do you think about this? How would you inform her of this decision?

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This doesn't seem like an especially unusual situation to me. You had a student for a fixed-term internship, during which time she pursued an angle that did not result in a publication. The student then moved on to other things, and so did you. You kept her informed of your work that is related to what she did as a professional courtesy, but she isn't interested in participating after the internship ends (nor does she have an obligation to). This kind of thing happens often, and nobody behaved badly here...
Do you have any indication that she expects to be a co-author on the eventual publication? (Corresponding with you about the work out of interest doesn't mean she expects to be an author.) She may not have any expectation of authorship credit.
If at some point she asks about authorship, I think you can just gently point out that her work did not lead to publication, and that you started working on the current approach (that did lead to publication) after she left.
If you do get the feeling that she expects authorship but she hasn't explicitly said so, you could say something to her like, "it has been interesting corresponding with you on this, X and I will be sure to thank you in the acknowledgement section of the eventual publication that he and I write." (If indeed you decide that you do want to do so.)

Good point, I have strictly no indication that she expects to author the paper, and I will clarify this with her.
Also, as for the situation being or not 'unusual', please check the additional details that I added about her thesis defense, which, I think, make the situation quite unusual.

A couple of things jumped out at me about the question that I don't think have been sufficiently addressed in the other answers:

This is a graduate student who was early enough in the program that she had not yet chosen her Ph.D. adviser when she was active on the project.
I deduce from some of the O.P's phrasing in the post and comments that the student's work was the application of a method or methods that did not work; this is based on these quotes: "I gave her something to do, she came up with different approaches and questions, wasted time, and then eventually did what I asked" and "she showed good motivation, but did not make any actual contribution to the progress of the project".

In my experience in academia, there are two general rules that are relevant here:

Graduate students should not be expected to have the requisite skills to be researchers yet--that's why they're here after all. But they should also not be expected to have a very good sense of where they belong in or how they should interact with academia generally. I don't mean here that they shouldn't be expected to be professional, but swing a dead cat in any major university medical center and you're likely to hit a young grad student who thinks that the best way to prove himself is to emulate Reviewer #3 at every lab meeting, talk, and defense. Most grad students grow out of this. A major, but rarely stated, part of grad school is learning how to interact with other academics, how to accept criticism, and how to be aware of the limits of your own expertise.
A researcher who tries a method that fails has not contributed nothing to the project: she has eliminated a hypothesis or found a null result.

In short I can only really recommend being cautious judging her, her attitude, or her ability to understand the limits of her expertise too harshly. In being naive/presumptuous/inefficient early in her career she is being nothing that grad students aren't expected to be, and the O.P. hired her knowing she was a student. My sense in similar situations I've been in during my career is that the marginal cost of putting a student in a middle-authorship position on a paper is usually much smaller than the moral hazard that I have underestimated their contribution or been predisposed to dislike it for other reasons (such as their attitude or implicit biases I hold).
Finally, I would suggest that the O.P. make it a teachable moment if the student truly has contributed nothing, especially given that she was hired as a student. Students are in labs as much for their own growth as for the labs growth, and assisting that growth is one of the responsibilities of being in academia.

Sorry, this is an undergraduate, not a graduate student, I clarified this in my post. In the academic system that I refer to, the internship is done before the beginning of the PhD. Also, please check the additional details that I added about her thesis defense.

Even given the additional information, I wouldn't be comfortable blessing her exclusion. The general advise I can give is that I hire undergraduates with the expectation that they may require more of my time than they save over their tenure. I don't expected to excel, though they occasionally do. I'm also not convinced that her behavior regarding her thesis defense is terribly relevant unless you think that it is okay to punish someone for not behaving well in one arena by refusing them credit for their work in another. I personally find that to be a morally hazardous policy as well.

@user16054 Agree 100%. Intellectual theft remains theft regardless of whether a thesis manuscript was sent late. OP sounds somewhat spiteful and approval-seeking, especially after the latest edits... I am so disappointed (but at the same time, not surprised).

You should have been clearer about the problem earlier on.  Several months ago you could have said that she needs to contribute to the calculations or else she can’t be a coauthor.  Removing someone from a paper this late in the process without making expectations clear is a bad situation for everyone.  Especially since she’s continued to engage conceptually with the project and hasn’t just disappeared.  I don’t trust myself to accurately gauge how important collaborators suggestions were, and wouldn’t feel comfortable removing a coauthor without their agreement outside of unusual circumstances.  If it were me I’d keep her on the paper, but not continue collaborating after that.  And in the future I’d be more clear in communicating expectations throughout the process.

This answer assumes that the student did perform her job duties.
In this case, be clear about the contributions to the paper. Roughly stated, "do this and that, and you'll get a coauthorship. If you do something else, it doesn't count".  Of course, you have to package the message in a polite way. (It's not easy, I know.) You could start with "I made a mistake. Earlier, I was not clear about the separation of our work for the continuation of project after your intership terminated. Let me make it clear now." After that, you might say that her "participation as a coauthor requires, as of now, doing verifiable calculations X, Y, and Z. Later more steps might get necessary from every contributing party; as the main author I will determine the nature of these steps.", for example. "Without such a contribution, I would have an utmost difficulty to include you in the paper." Moreover, motivate according to @aparente001.
 It's a bit late to write a clear statement now, but better late than never: postponing makes the matter worse.

@James My first line is an assumption. Earlier, I was under the impression that she did not perform her job duties. Then, you said "on a purely formal level, she has done what she was asked to." Now, I guess she was not paid for doing her dissertation properly, but for her project. The who diss matters seem unrelated to me. However, since you still have doubts, I'll restore my previous answer with the negated assumption.

I have definitely worked with undergraduates who enthusiastically tried my ideas which have failed, and then with my own hands and brain I have been able to get the same project "working". In this case if I think the student did the negative experiments correctly and diligently the negative results they have obtained have still been crucial to project development and I have offered coauthorship to the student when the manuscript is in its final form.
In this case it seems like you have personal problems with the student and you feel their work quality while contributing to the project was of poor quality. I do not think you should be required to offer coauthorship to the student. However, by asking this question on stackexchange I pick up on a worry that the student does consider themselves a coauthor. You may want to protect yourself by sending a written copy of the manuscript to the student that specifically describe their contributions in the acknowledgements section. That way you would have a written record of manuscript approval. If you go through the submission process without resolving some grey areas you run a risk of going through an authorship dispute after the manuscript is in review or is published, which is something you deeply want to avoid.
Of course you are not PI, just first author. While you have some power and discretion in this situation, you do not have final authority. Hopefully you can also communicate openly about these issues with your boss.

I support in broad terms the short answer contributed by @Hexal, but there is a less abrasive, more positive way of accomplishing the same thing... even if you are acting alone.
But you needn't act alone.  You can talk to her advisor!  Her advisor can help her understand your conundrum, and what she would need to do in order to be a co-author.  Her advisor is in a sense her ombudsman.  Give him or her a chance to function in that capacity.
The more positive approach, which you might not need to carry out, on your own:

emphasize the positive, e.g. her emails show that she is actively thinking about the project.  Before reading her the riot act, acknowledge her unexpected continued interest.  I realize it's presumptuous of her, but at this stage, that's not a criticism that needs to be aired with her.
Start out by dangling a carrot -- a possible co-authorship.  Explain that a co-authorship might be possible, IF she is willing and able to make a significant contribution to the project.
Now you have her attention.  Drive the thing through to the end now, by describing enthusiastically a task (or set of tasks) that are needed in this new phase, and pick out some characteristics of hers that lead you to think that task would be right up her alley.  This is a strength-based invitation to collaborate.
Acknowledge her interest and enthusiasm.  Then explain that in experimental work, creativity is hugely swamped by willingness to do grunt work.

Thanks. I would be happy to be proven wrong, but, frankly, her new advisor typically wishes her to do work only for that new advisor, not for the old one!!! Exceptions do happen (and I know them), but, IMHO, they remain exceptions. So, you would speak to the new adivsor probably only if you wish her to stop being part of the project.

@Hexal - Are you talking about the OP's former intern, based on the question, and OP's comments?  Or are you talking about academia in general?

Since there is almost no information regarding her new advisor, I assume the typical general case.

In addition to being her ombudsman, her advisor is also her -- well, advisor.  That is, her advisor can help her make these decisions, and also guide her to grow into a more effective collaborator.  I would not assume that her advisor would reject OP's proposal without thinking about it.  Most importantly, OP's former intern should not make decisions about the project without consulting her advisor.

@Hexal - I don't understand "directions of the decision."  Also, why are you talking about them being detrimental to the new advisor?  I didn't say that in my answer or in a comment.

@Hexal - I still don't understand your comment.

The advisor has a choice of how to advise the student. In cannot imagine the situation in which the advice of the advisor would be detrimental to the advisor. In general, I cannot imagine the situation in which the advisor would give the time of his/her student to someone else. Note that the student is  collaborating "effectively" (for the student) already now - it is simply not useful.

My advisor always advised me in terms of my best interests.  He never advised me according to his best interests if they were in opposition to mine.

In general, it's in your best interest to support the current advisor. (Exceptions happen, of course.)

@Hexal - I do not understand where you are going with this at all.  Who's talking about supporting or not supporting the current advisor?

Do you wish to say that the student does not support her current PhD advisor in any way? That would be wrong: her success on her thesis is a contribution to the success of her advisor (regardless of whether she is paid for some work for him or gets funding from elsewhere).

@Hexal - Do you mean, am I observing that she doesn't?  Or do you mean, am I proposing that she not?  I wish you would just lay out your point of view and your reasoning.  I CANNOT understand what you have been trying to get at.

I'm sorry, I really don't get your question. Perhaps, its me who CANNOT understand you, not the otherway round. "She doesn't" do WHAT?

This answer assumes that the student did not perform her job duties.
she did not make any actual contribution to the project - forget her. I am extremely surprised that you even ask about it.
The fact that she showed motivation does not count. What counts is what she did. If (and only if) she does her job, she gets the money. If (and only if) she does a good publishable job, she gets scientific credit in addition.  That's how it should be. None if these seems to be the case.
So, if she ever writes an e-mail, this would mean that she still wants some of the credit. In this case, fight back, exercising a bit of pressure, getting an o.k. from the colleague first. One approach could go in the following tone:

Thank you for your e-mail.  Please notice that you did not fulfil your job duties, but you were paid.  There have been no real contributions to the project from your side.  As for high-level ideas, these are outside of your scope.  Necessary calculations have not been done, despite my several written requests to you, namely on ... , ... , and ..., which was still before your project ended.  This lack of contribution is not how it is supposed to be.  Therefore, we would appreciate it if you return the salary you received for the internship.  Please transfer USD ... to the account ...  until ... .  If we don't see a reverse payment, we feel necessitated to undertake legal steps. Thank you for your cooperation.

Of course, adapt the text accordingly: you will get the exact wording better than me since you know the details. If your institution has a legal department, you might even delegate the exact wording right away to them. Make sure that the letter goes via snail mail and has an institution stamp.  Use the account number for that project.  Now, guess, would she ever think of claiming any academic credit after a letter in such a tone?

Do not not how it works all around the world but I know enough places where this is not the way to do. And it lacks legal bases, too.

@Alchimista That's why I pointed to the legal department. They know the legal basis better. Note that the student is unlikely to know the legal basis. Still, before it comes to the legal matters, you have to express your claims, and that's what the letter does.

Sorry.  Is non sense legal bases or not. Beside I personally totally dislike such a potential behaviour,  it practically means to forget enrolling students for a good while.

I would also use much nicer tones in an eventual letter.

@Alchimista Not all students do not know each other. In a global world, this letter would go unnoticed. You would get good students if you are good, and you are good if you can do good research and force the others to do good research.

The last is totally another point . Which I even mentioned in comments to the Q

I'm confused. Where does the OP say that the student didn't fulfill job requirements during the internship? It's not unusual for a student on a short term internship to try out an approach that doesn't work out, that doesn't mean they didn't do their job. The student is also under no obligation to keep participating in the project (e.g. checking the OP's calculations) after the internship has ended.

@ff524 That's why the exact wording should be done by the OP. As for after the project, my letters deliberately says nothing about it.

You want to ask her to pay you back ???? In what world is this appropriate ? In absolutely no situation is it relvant to ask anyone to pay back a salary. OP also states that it is "definitely fair to include her in the acknowledgments". Why would you send such a hateful message ? Such a practise is also totally illegal in several countries.

@everyone It's inappropriate to ignore one's job duties. Moreover, the message is not hateful. It uses no derogatory language or the like. It's a business letter.

@Hexal If a student under my supervision doesn't fulfil what I perceive to be their duties, I as the supervisor am to blame, not they as the supervisee. If they were neglecting their duties to the point where it is breach of contract, I would certainly have to inform them about that during their internship. The way I read the OP, the student did what she could, and did what she was asked to, but made no relevant contribution to the project itself.

@sgf did what she was asked to - that's your assumption. I assume the contrary based upon OP. But, again, I'd be happy to wait for the clarification of the OPer. You might be right, after all. Hypothetically ;-).

One thing I agree with in this post: Please do defer to your university's legal department before considering to mail off one of these letters...

Probably the worst answer in all of Stack Exchange. Extraordinarily awful.

@Hexal as unfortunate as the situations sounds and as appaling the student's attitude is described (I've seen similar things, they do happen and annoy me greatly), uttering empty threats is at least an unneccesary escalation and maybe even dangerous for OP (depending on whether the tone is perceived as hostile or even threatening). A salary that's been paid is very hard to claw back!


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Can a conference paper with all weak reject get accepted at the end?
Those who have experience in setting in conference program committees: can a paper with all "weak reject" reviews get accepted for publication at the end?

If they don't have enough papers, then yes. This is more typical of some workshops. Good conferences mostly have much more papers submitted than time slots for talks. This is what defines the acceptance rate and why it is an important metric.

In case you refer to a proceeding it would be better no to publish. Better go through what caused the weak rejection. I add this because in my field submitters are never aware of the referee procedure for their work to get a slot as talk or poster.  This is just after conference, for the corresponding proceeding manuscript.

If nobody wants to accept the paper, it shouldn't get accepted.  This holds even for workshops and symposia with very loose standards, as the reviewers should be making their judgements based on those standards (e.g., "this paper is pretty half-baked and poorly written, but we're just looking for interesting ideas to discuss, so it's a weak accept").
Thus, the answer ought to be no, it cannot be accepted (except in certain unusual cases, such as the organizers deciding there is a hidden nugget with value enough for shepherding to improve the paper).
A struggling, eccentric, or fraudulent organization, however might just not care, and accept it anyway. If you have this happen to a paper of yours, you may wish to reconsider whether you really want to publish it in that venue.

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Is a bachelor in Computer Science Engineering a viable idea if my ultimate objective is to be a mathematical modelling engineer?
I want to mathematically model reality and use these models to solve problems. Will a degree in Computer Science Engineering allow me to do this?
Additional background. Reading the overview of a master's graduate degree (MSC to be exact) in modelling mechanical engineer, it really spiked my interest. I am not a mathematical genius though, and I need to at least see applications and more or less concrete problems/implementations. Also relevant could be that I am a senior in high school and I learned programming all by myself.
Systems I would find interesting to get to know better by creating mathematical models
I am most interested in physics and general mathematical patterns, like the approach found in Albert-László Barabási's book Network Science.
I am not interested in specializing in any particular field just because of the field itself.

I couldn't really understand the question and I've edited in an attempt to make it clearer. (OP: Please edit further if necessary.)

I don't know what a mathematical modeling engineer is, but "modeling reality using mathematics" sounds like graphics programming / computer vision (which is a subfield of CS) might be for you.

I mean mathematical modelling

@user7474 I didn't mean the spelling, but the concept behind it.

Of course, me too
I meant not visual modeling/modelling I really don't know, but mathematical representation/approach

@user7474 Adding some concrete examples of what you want to model might be useful. E.g., biological systems, civil engineering systems, ...

J.Ari made a good point for you when you asked the same question here : https://engineering.stackexchange.com/q/18937/10902

Yes, I found that constructive as well, but I am not sure I am up for that much of raw mathematics for that long a time.

From my own experience, Im doing PhD in computer graphics, mainly physics simulation, and I have a master in mathematics, the program was called "mathematical modeling" it was mainly analysis, partial differential equations, numerical analysis and continuum mechanics.
Physics simulation is nothing else then solving partial differential equations and I can see that a math background is much more useful then computer science background when I look at my peers.
So I would recomend a math degree with a focus on numerics and physics.

edit: I see that you are a little bit worried to study math, that it might be way too much math for your taste. However, you are in a great position because you have a great motivation to study mathematics. You can always come up with things to model with math you are being thaught at school. From using projective spaces in computer graphics to fourier analysis in image processing . BUT remember one think. Do not expect that profesors will motivate you(it is a bonus if they do) but you have to activelly look for things that motivate you, everyone gets excited by different things.

Another point to consider is that the kind of math that a computer scientist needs is very different from the one a mathematical modeling expert uses. CS people mostly want discrete mathematics (combinatorics, some number theory, sequences, ...), modelers need continuous mathematics (differential equations, function spaces...).

@tom, thank you for your detailed answer and also for adding an edit, I find them really useful. I know I can't expect my professors to motivate me, and I do want to participate in projects and researches related to my interests

@FedericoPoloni yes, that is one thing I myself have been wondering about, and I know I have to take that into consideration. I think I want both, at least for the time being.

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How to "tailor" the statement of purpose for a particular grad school
A prospective graduate student is normally not hyperspecialized and inflexibly hyperfocussed on a certain well-developed research line (like a more mature mid-career researcher could be) just yet.
That is to say, even if such student worked on topic "X" (say, in the area of algebraic geometry) for their master's thesis, they may well be flexible and willing to expand their horizon and start working on any interesting topic in algebraic geometry that falls within a certain wide range "A" (that possibly includes "X").
So, I assume, any school which is (1) reasonably prestigious, (2) conveniently located, (3) endowed with "good" mathematicians that work on a quite wide range "A" of topics within algebraic geometry is in principle a good fit for an application of said student. Any other reasons are (at least in my opinion) less relevant.

In this context, what does it mean to tailor a statement of purpose for a specific school?

In particular, how can one convincingly and professionally show that a certain school is a great fit for their academic career and education?
I have heard that the "Why is our university a good fit for you?" question for a statement of purpose is just a way of checking if a student has "done their homework". But what is that actually supposed to mean?

Niggly point: I think you mean "prospective graduate student" (only pointing this out as the rest of your thing is so well written).

I have been involved in graduate admissions for a quite a few years. Answer below is based on that experience.
This question about fit is a way to make sure that you know approximately what you are getting yourself into.
You do not have to stroke professors' egos by telling how great they are. It does not help your application. If you have interest or past experience in some area, mention it of course, but do not engage in gratuitous flattery. Instead we want to know why this school is suitable to you at all. For example, if you are interested in doing mostly algebraic geometry, but are applying to a department that has no algebraic geometers, this would be a poor fit. If there is a reason you like the location (e.g. a significant other working nearby, or cheap flights home), it does not hurt saying so either.
What we want to avoid is to have a 3rd or 4th year graduate student drop out because they cannot find an advisor in their preferred area, and are utterly unhappy because of location or because of something else.
In general, the statement of purpose is your opportunity to present evidence that you will make a good researcher one day. That includes your scientific background, various personal traits as demonstrated by your past, and your accomplishments. See an excellent answer to another question by JeffE for more details.

+1 but I should say that for universities that imagine themselves in the top five of their disciplines, mentioning the location as the sole reason you want to attend would be insulting at best. It’s a plus for mid-tier universities though.

@RoboKaren Did you call my university mid-tier? :-)

This particular question's answer can be found in the job advertisement.
I usually prefer writing in a Question-Answer fashion. When the position is advertised, there will be a list of required skills. For each requirement, stating how you fulfill that requirement is the most straightforward and neat way to answer your question.
As a side note, you may include the possible reason why the committee would not want to pick you, and justify your counter argument.

Thanks for that side note. :) How much do you think that "makes or breaks" your application? I suppose it could backfire if the reason given by the student doesn't prove to be as convincing as the committee's own understanding of the candidate would have been.

Chances are, the committee can guess everything that the applicant guess, and even more. Even if that is not the case, they will appreciate that you know your weaknesses. Of course, I assume that you write them what is already obvious from your CV :)

Speaking as somebody who's been writing my applications for PhD programmes recently, here's what I did (whether it's "convincing" or not is yet to be seen!):
My statement of purpose at its core outlined my (relevant) experiences as an undergraduate. For me, these included my degree, my final year project and my year in work experience during my degree. These core things didn't change much from application to application.
What did change is how I related these to where I was applying/ to which department I was applying. For example, one of the universities I applied to is not in the UK, it's abroad. I then made more of an effort to highlight things I've done where I had to work in new environments, and tackle new problems.
Depending on the supervisors/projects I was going for, I might highlight units I've studied which relate to those particular things, and why they gave me an interest in the project I was applying to.
As opposed to vastly changing the content, I changed what I put emphasis on. Depending on where you're applying, some experiences/units you've taken/etc will be more relevant to some places (or to some particular projects) than others.

For your starting default draft, write it in a way that will convey your excitement about diving into research.  It shouldn't look like a CV.  You can show motivation, creativity, fantasy and enthusiasm.  (No one is going to invoke a money-back guarantee several years from now if you didn't realize your dream as you described it in your statement of purpose.)
Now for the tailoring to the specific school.

Look at some webpages for the university and the department.  Try to get a feel for what makes this university and this department different from the average, and what appeals to you.  Check your draft to see if there's something you can emphasize, to convey this.  Example 1: if the university emphasizes early student involvement in research, and you participated in a research project as an undergraduate, make sure you've talked about that experience, and how it shaped your development.  Example 2: if the university has an innovative interdisciplinary focus, you can use that as a hook.
Which of the department's specialty areas intrigue you?  Read some faculty bios.  When editing your essay, see if you can tie some of your fantasy ideas in with something you read about in the faculty research interest overviews.

The university want to avoid a situation where they take on a student who fails to complete - look at your application from their point of view.  They want to know that you will be competent and happy enough to survive four arduous years at their institution, with the loneliness and destitution that is the life of a graduate student.  Give them the signals that show them that you will survive.
Your statement of purposes is an opportunity to show that you have the diligence, foresight and skill to check the research strengths of their school, and you have some sensible reason for regarding that school as a preference for you.  The university will want to know that your interests are broad enough to allow you to find a suitable research topic that can be supervised by someone in their faculty (preferably with backup, in case they have staff changes during your candidature), and you are able to narrow your focus when needed to get down dirty and do the work.  They want to know that you have interest backed up by competence and diligence.
Demonstrate some knowledge of the faculty and their research interests/projects: This is an opportunity to demonstrate that you have done your due diligence and checked the various school webpages, faculty profiles, etc., to show that you are aware of the broad research strengths of the school, what they are best known for, and how reputable their research is in different fields.  If you have read some of the papers by their faculty, mention this.  If you have seen some of the faculty give talks, presentations, etc., in an area of interest to you, mention this.  Show that you have a broad awareness of the main strengths of the school.  If it is a prestigious school, their strengths are probably numerous, but you should still be able to mention a few specific areas.
Mention some areas of interest to you that align with the school: It is important to be able to mention at least a few research areas of interest to you, and be able to point to faculty/research groups in the school that work in those areas.  Bonus points for being able to point to some specific work by the faculty that you particularly liked and is of interest to you (bearing in mind that you are only just applying for grad school, so you probably don't have a wide knowledge of the literature in any area).  Show that when you have learned about various fields in your undergraduate degree, it gave you lots of ideas for possible avenues of research.  Ideally you will have lots of ideas -your mind is flooded with them- and you can't wait to talk to the professors at this school to see if any of your ideas are feasible!
Back up your interests with your skills/qualifications: If you're interested in a particular field of study done by this school, what evidence can you show to demonstrate that you are well-qualified to start study in that area.  Did you do well studying this area as an undergraduate?  Did you get good grades, win an award, write a good project, etc.?  Did you ever do any research work on it as an undergraduate?  (Remember that most of the applicants to a prestigious grad program probably nailed their undergraduate degree, so make your achievements relative to this cohort.)  Ideally you want your statement of purpose to show interest backed up by evidence of high competence (relative to other applicants).
Show how excited you are to study there: Convince the school that you will be happy there, and you have the personal skills to make friends and create a life for yourself that will sustain your time there.  Unhappiness/loneliness is a killer of graduate degrees, so they want to know that you are a low-risk candidate for dropping out.  You should have a tone of excited anticipation - show that you are looking forward to graduate school, and you are excited about studying there.

In my experience, research interests can be even more flexible and broad than you suggest. In particular, it is extremely common for your research interests as a graduate student to change during the first few years, even drastically -- and everyone who is reading your personal statement knows this.
Let's say you write on your application, "I am very interested in topic Y and more broadly, area B". This does not imply that you have research experience in topic Y or area B; it implies only that you are interested in it (and have a passable understanding of what that topic or area entails). It will certainly look favorable if you do have experience in Y or B, but what is most important is that you have any experience, even in a completely different area A.
So what does it mean to "tailor" the statement of purpose to a particular university? It essentially means, (1) choose your interests wisely, and (2) don't be afraid to pick interests outside of what you are normally comfortable with, if they will make your application look like a good fit. I especially emphasize (2) -- it is very easy to think that you cannot express interest in something unless you have studied it extensively, but this is not true, and anyway, how will the admissions committee be able to tell? As long as you get someone to read over your application and it sounds like you know what you are talking about, your interests will be taken at face value.
A corollary to this is to make sure to justify your extensive knowledge and research experience separately from your interests. Since people fake interests all the time, just saying you're interested in a topic doesn't imply experience and knowledge, so you should make sure that experience and knowledge are sufficiently conveyed in one way or another.

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text:
Do all accepted students in top-tier engineering programs get invited to interview before getting admitted?
I'm an (international) electrical engineering applicant applied to some top-10 graduate programs in US (say, MIT, Caltech, UCB, etc.). As I check the entries of "thegradcafe.com" every day associated with the programs I've applied to, there are many entries which declare interviews with some POIs. So far, I have received no interview requests from any target school. Thus, I'm a little bit both curious about the importance of interview and anxious about the interpretation of such a lack of interviews in my case. To be specific, are all admitted students to top-tier engineering programs invited to interview before getting admitted? In other words, should I expect to be rejected if no one invites me to any sort of interview?
PS. If it helps to clear the situation, I'm currently an M.A.Sc student at a well-known Canadian university. So, both my university and my supervisor are pretty famous to the research community of my field.

university of toronto is surely very famous... u know that MIT etc are on a different level?

@NostradamusJR No need for snobbery here. For any given field, there are a good number of institutions in the international network of top schools. The distinction at that point tends to be less about schools and more about people within a community.

@JeffE My research background is actually in control and robotics; however, considering my recent research experiences in my master, I've announced my interest in bimolecular feedback systems.

@JeffE Given the response, could you convert your comment into an answer?

Not all graduate programs in the US have (official) interviews at all.  In particular, except for Caltech, I'm not aware of any US computer science graduate programs that regularly interviews applicants before making admissions decisions.
There are a couple of sporadic exceptions, which are almost always handled by phone or skype:

Departments sometimes interview non-domestic students to check for English fluency, especially if their TOEFL (or equivalent) scores are missing or borderline but the application is otherwise strong.
Some individual faculty arrange informal interviews with prospective students before deciding whether to offer funding.

Outside computer science, some US engineering departments do regularly invite PhD students for on-campus interviews before admission, but I think this is more the exception than the rule.  In particular, as far as I know, none of the engineering departments at my university (which is highly ranked in most engineering disciplines) do this.

After one is accepted to a USA engineering graduate program, it is very common to be invited on "post-acceptance recruitment trips". The purpose of these is to recruit you to accepting that university's offer and begin making connections with prospective faculty advisers. However you should also treat these events as a type of interview in that if you behave inappropriately the offer of support could still be withdrawn and indicating a lack of personal commitment to research could alienate potential faculty advisers from you. For example alcohol (!) could be provided to you in an informal setting and your ability to act professionally with your peers will be judged by current students and faculty.
If you have been accepted to these departments with stipend support, feel free to contact the department's graduate program coordinator and ask specifically about the schedule for recruitment trips.

But the records of "thegradcofe.com" indicate something different. Specifically, many posters have an interview with a PI and then they receive an informal email from their PI in addition to a formal acceptance from the corresponding university. Put differently, I'm wondering whether or not interviews are pre-steps of admissions, not a post-step one. Additionally, the aforesaid interviews are mostly by either phone or skype, and there is often no recruitment trip even for American (domestic) applicants.


text:
Professional ways of saying "Keep me in the loop in case you have some other projects I may be able to contribute to"
I’m a student and I’ve been visiting a collaborator of my advisor to work on a project. The visiting period ends in a few days and it has been productive and enjoyable. We have worked well together, but the professor has no shortage of students or collaborators.
What is a more “refined and professional” way (for lack of a better term) to phrase the following message:

Keep me in the loop in case you have some other projects I may be able to contribute to.

You can weave it into a concise thank-you note.  But most importantly say it to the professor face to face (both the thanks and the eagerness to come back in the future).  Also, let your advisor at home know how well it went and how much you'd like to do it again in future.

Or just say it simple and straight ... I really enjoyed this project, would love to do it again if there is any possibility ... personally don't see reason to complicate or over-think it to much.

Maybe it is just cultural, but I would say most of the answers here are informal, which to my mind is not what I would call professional and refined. Depends what you are looking for.

I really enjoy working with you.  I hope you'll think of me if an
  opportunity arises where I might contribute to one of your projects.

Thank you. This is just the kind of phrasing I was looking for.

"Keep me in the loop" sounds fine to me.  It's a pretty standard expression nowadays, and academia typically isn't hyper-formal anyway.  I would just add "please".

Thanks for your answer. I actually wasn't questioning the idiom, but the whole message, which I'm afraid may give the "cold call" vibe.

I really loved working with you. It has been very productive for me. I would love to be able to do it again. Thank you.
You are this way letting him/her know how much you liked working together, you valued it (it was productive) and you would love to do it again. You are not asking for anything. You are not saying "keep me in the loop" that puts the burden on him/her, or anything like that ("let me know ...", "think of me ...", etc.)
You are making your point clear, no need to say anything else, and no pressure whatsoever on him/her.

That's a thank you note, which doesn't seem to address the OP's question.

If you read the whole answer, it is exactly an answer to the OP's question

I have read the whole answer. You answer your own question, rather than the one asked by the OP. I appreciate that you might believe the OP has asked the wrong question, but I disagree: I believe a valid question has been asked.

@user2768 I never said the OP has asked the wrong question, but you can believe what you like.

I never said that "[you] said the OP has asked the wrong question." I did say that you've answered your own question. In particular, the OP questions how to phrase a "keep me in the loop" message and you dismiss this when you write: "You are not saying 'keep me in the loop'..." Perhaps you can clarify how your answer relates to the OP's question?

I think it is useless to keep answering you.

I'm sorry that you feel that way.

TL;DR

I really enjoy working with you and if possible I would appreciate the chance to contribute to one of your future projects.
  Would you mind if we stay in contact?

(The answer might be Influenced  by culture and heavily depending on the hierarchy of your field.)
If I would receive a letter (or similar) containing a statement like:

I really enjoy working with you. I hope you'll think of me if an opportunity arises where I might contribute to one of your projects.

I would immediately wonder why you plan to maintain a passive role in this (as I stated before might be cultural), since it essentially puts multiple burden on me:

I need to keep you in mind (besides all the other stuff)
I need to judge whether certain topics would fit.
I need to initiate the conversation.

Why don’t you assume a more active role in this interaction.
Not being a native (English) speaker, I am not sure how to phrase it properly (please edit in case).
I would love to read something like the following:

I really enjoy working with you and if possible I would appreciate the chance to contribute to one of your future projects.
  Would you mind if we stay in contact?

I see your point, but "keep me in the loop" means "please contact me", not "may I contact you?".

Right...hm...I guess I did not make myself clear enough than. My point is that I think it would be more professional not to ask to be kept in the loop, but to take action rather than waiting to be contacted and by this defer the responsibility.

I'm not sure about what you are suggesting: 1) call him periodically and basically say "what do you have for me today?" 2) contact him if I want to pick his brain about my own ongoing work. I'm surely going to do (2), but I think (1) is a bit unprofessional, although it surely is more pushy (and possibly effective) than "I really enjoy working with you. I hope you'll think of me if an opportunity arises where I might contribute to one of your projects."

You are right @Dal; 1) is currently pushy (that’s why I requested a native speaker to edit it) but it also would show pro-active effort. I’ll try to edit

I would like to work with you again. Please do not hesitate to contact me if you see an opportunity for our further cooperation..

The OP is a student. He wants to contribute to other project the professor has. I can hardly call it "cooperation".

As noted by aparente001 and discussed by others:

You can weave ["Keep me in the loop"] into a concise thank-you note.

But, keep me in the loop is rather vague and leaves your collaborators unsure of what you are offering. Be more precise: mention what you're willing to contribute. E.g., additional experiments, writing-up, proofreading, ...

The OP commented:

I actually wasn't questioning the ["Keep me in the loop"] idiom, but the whole message, which I'm afraid may give the "cold call" vibe.

I'm uncertain what the "cold call" vibe is, I think you mean: "Keep me in the loop" lacks sincerity. My answer is intended to address that.


text:
Reality check on advice for statement of purpose
I'm about to write a statement of purpose for graduate admission and my advisor passed along to me the following pieces of advice to keep in mind while writing  (based on his style and experience in admission commitees).
I completely agree with every point he has made. However, I'd like to have further insight and suggestions and yet another reality check on the soundness of these pieces of advice.

Advice from my advisor:

Don't try to sell me my own research area by explaining to me how fascinating it is. Do show your interest by showcasing your previous
  experience working on (or studying) related topics (not necessarily
  very closely related).
Don't tell me how amazing, prestigious, perfect my institution or research group is. I know more than you do about that. Also, chances
  are that you are applying to a lot of schools and don't have
  inflexible interests yet, which is fine. Just tell me that you may be
  willing to work with a certain research group that seems more or less
  in line with your previous experience or future directions.
Don't tell me a cute story about how you fell in love with my field when you were a kid (or high school senior or college freshmen). I
  just don't care. I want to hire a soon-to-be professional to join my
  research group, not a little boy with a cute story. I only need to
  know factual information that show your commitment to the field (for
  example, courses/conferences attended, projects undertaken).
Ban ill-crafted, pseudo-literary, flowery, cheesy narrations. Just  write facts. Straightforwardly. Succintly. Accurately. I'll draw conclusions for myself.
Ban any buzzwords, meaningless adjectives and adverbs. If they don't add concrete information, but are there just an ill-advised attempt
  to impress, cut them!

I wished every applicant read this list and stuck to it...

I would add some advices regarding the presentation.  Spell-check, Be consistent in your formatting, Make the structure is easy to follow, and, as Nicole suggests, try to be future-oriented.

I agree with your advisor, especially about the second item on the list. When I was in charge of graduate admissions, I got to the point where I wanted to automatically reject anyone whose statement of purpose contained the word "esteemed". (I never quite had the courage to implement that policy; I think all those applicants got rejected for other reasons.)

All good advice.  In addition, may I suggest that a statement of purpose should answer the question, what is your purpose in wanting to do this? What will you do if they admit you?  It should be future-oriented.  What are your goals and how does this fit into your plans for getting there?

My research adviser and my academic graduate adviser at my undergrad aerospace engineering department provided similar advice as your professor when I applied to grad schools. They dread reading the same "I fell in love with airplanes as a child" opener. Their advice prompted me to write a formulaic but clear statement of purpose that I think served as a breath of fresh air for one my successful "dream program" applications.
My SOP was 1.5 justified alignment (do this no matter what you write, so much neater) pages with clearly labeled sections:

Motivation/background, 1/4 page, 1 paragraph:  How a boring internship encouraged me to pursue graduate studies if I wanted a challenging career. This is where you could insert a thesis statement or an actual statement of purpose.
Relevant experience, 3/4 page, 4 paragraphs: lab experience, publications, etc. This is where you show off that you are a professional student as your prof explained.
Career goals, 1/4 page, 1 paragraph: what do you want to do after school? How will a grad degree help? This is where you can weave in Nicole Hamilton's suggestion of your short- and long-term plans.
Institution selection, 1/4 page, 1 paragraph: Why are you applying to this department? Why are you applying to this school?
Funding, 1/4 page, 1 paragraph: How will you would pay for grad school if you don't get a paid RA position? (TA, private tutoring, external fellowships, etc.).

You may have more or less information for each section, but I would recommend this structured and straightforward style. I think anything more than 2 pages is overkill though (except for NSF).
In summary, make reading your SOP as easy and least-cringe-y as possible.

I'm a current graduate school applicant. I included a short story in my statement of purpose to give a few details of my background and ultimately show how I overcame barriers to be a successful first-generation college student, like receiving a full-ride scholarship based on my status.
To modify point 3, telling a small story about a relevant part of your background to highlight some characteristic of your personality, like dedication, can boost your application. This can include describing overcoming hardships as a first-generation student or a foreigner. Plus, it will add some "personality" or "insight" to your application that can't really be described anywhere else.

Reasons this is generally sound advice (and what's missing)
Professors must read many statements of purpose, so following this advice will make it easy for them to get the info they are trying to find.  Honestly, the stories about when a student fell in love with a topic aren't helpful in determining if they will be a productive scholar (and get really boring after reading 20 variations on the same theme).  Sticking to the facts will help the reader get the information they need quickly and prevent your letter from being tossed aside.
Additionally, (depending on the specific field) scientific writing tends to be focused on communicating clearly and precisely; good scientific writing leaves out unhelpful digressions and imprecise terms.  Following the OP's advisor's advice will tend to make a statement of purpose look more like good scientific writing.
What's missing
I agree with the advice that it is unhelpful to tell a potential advisor that their research group is prestigious/perfect.  However, as the first point of advice implies, it is critical to communicate your "fit" with the work being done there.  Potential advisors want to know whether you understand what you would be getting into, and that the work is a good fit for your interests and skills.
This could mean that you have already worked on related projects, but not necessarily.  For instance, I once read a statement of purpose (and accompanying letters of recommendation) that emphasized that the student was not afraid of teaching herself new skills and learning new topics without hands-on guidance.  This sort of independence is critical in my area of work (which is interdisciplinary and requires understanding work coming out of multiple fields).
If your key strengths would make you a good fit for the type of science being done in that group, you must emphasize this.
Ultimately, as others have pointed out, professors are interested in hiring potential junior colleagues who will be productive.  It is OK to deviate from the specific points in the OP's advisor's advice, so long as you (clearly and succinctly) speak to this main point.

I see your advisor's point of view, and it is safe to say that he is right. However, there are exaggerations which might lead to a nonsensical cover letter. The tone in these advices seem a bit disrespectful to candidates as well.
If you stick to this advice 100% this is what you should write,

To Whom It May Concern,
  I am writing this letter to declare my
  interest in the open position a the research team ABC. I have studied on
  this topic for two years, and I have four publications. I am willing
  to work with people from different cultures and countries. This
  research area also fits my future plans. I have attended courses X, Y
  and Z, and passed them all with good grades.

Set aside this information can be obtained by only looking at your CV, the cover letter does not reveal any personal information.
Unfortunately, by giving an advice of not using ill-crafted, pseudo-literary, flowery, cheesy words, ironically, he himself uses some buzzwords, meaningless adjectives and adverbs that we hear in Hollywood movies, said by the "tough" commander or detective.
The research team that you are applying to is looking for a human being to work with. And chances are, they want to have a wild guess what kind of person you are, before moving onto the second stage.
Explaining why you want to work in that research direction, and how is it going to contribute to the literature is not advertising and "selling" a research area.  It is you telling the committee why they should think that you are more ambitious than the other candidates.
Also, people usually want to explain why they are applying to a certain position. This is not a cute story. This is called motivation. By same reasoning, we might also drop the cute 5-year-old-boy stories from scientific papers, and get straight to the business.
Therefore, I would not 100% stick to these advices, and also not ignore them:

Your background and ambitions are definitely needed in a cover letter.
Stating your aims and goals during your employment is very important. More important is to convince the committee that you can achieve a satisfactory amount of them.
Why do you want to work in that particular research group? Why is that research area important for you?
Consistent formatting is important in terms of readability.
Remember to explain that you can fulfill all the requirements in the job advertisement. Preferably in a FAQ fashion, first, requirement, than your qualification for that requirement.

Remember, cover letter is the first step that a committee gets to know you, and it should be treated that way. If you cut out all personal details, then better is not sending the letter at all. Just a CV would be sufficient.

-1 for the initial criticism. Banning flowery language does not mean banning evidence.

The criticism ia not about banning the flowery language, but the tone of the advice. "Do not use flowery language" is different than "I do not want a 5-year-old whiney promary school baby boy who uses cheesy, unnecessary, flowery narrations." This is just a research team, not the marine.

-1 for the given reductio-ad-absurdum example not understanding the core question. The reduced example is an inquiry to a faculty member, not a statement of purpose for a graduate school application. Different documents, different contexts, different expectations (and the faculty inquiry letter should be very short and direct).

Tiny modifications can easily convert it to a SOP. e.g. "To Whom It May Concern" instead of "Dear Prof," and "the research team" instead of "your research team." And this actually proves my point.

Nothing you have written "proves your point". Your characterization of what OP should write "If you stick to this advice 100%" completely mischaracterizes the advice.  The advice is not itself a statement of purpose, and therefore does not need to have—and arguably should not have—the same tone as a statement of purpose.

Also, a PhD position is a job. In any other job, nobody will hire you because "you have always been in love with quantum mechanics", but because you have a generally solid, thorough understanding of the field and are able to draw motivation from the field itself, rather than from your childhood or undergrad memories. If you have some research experience, it should be very easy to write this part professionally. If you had experience but were not involved, it's very easy to spot it when reading a motivation.

I think I made it not very clear that my main objection is about the tone, and exaggerated expressions. I do agree that one should not include how they "fell in love" with some topic, but the motivation behind studying the topic is needed in a statement of purpose.

I think the issue you are getting at here is the danger of being "overcoached" by focusing too much on avoiding mistakes. To succeed one needs to take risks.

+1 to this answer. I agree about the tone of the advice and the result that would occur from an overly literal interpretation of the advice. In any kind of writing you are always balancing conflicting objectives and trying to find an optimal balance. I think the professor's original comments are meant as, essentially, "tone it down and be more objective" in response to typical statements. But that doesn't mean you can't go too far the other way. I think this answer provides some useful counterpoints that should also be kept in mind.

I want to add a note I don't see in the other answers:
The number one most valuable piece of advice for any writing of any kind is the following: get others to read it and give you feedback. Ideally someone like a professor who is interested in helping you, but anyone is better than no one at all. Your friend or mother can still tell you if your statements are confusing to follow or lack a clear point.

Overall I think there's good advice here, but I want to provide the opposing perspective on two points to help you "thread the needle."

While you don't want to explain to an expert why their own field is fascinating, it is an important skill and impressive in a candidate if you can explain in objective terms why the field is important and how it fits into a broader context. This shows you have done reading outside of any research projects you have done and have a feel for the big picture.

One reason the graduate school statements sound the same is that you don't yet have a deep background of work to show your interest in a topic, which is fine! I think people compensate for this background, by trying to use flowery language to convey their interest. Instead, I think the better route is to be concrete. Think about ways to connect the work you have done (even if this is "just" course work) with the work the group you are interested in is doing ("I took advanced course X which will prepare me to study topic Y in more detail in grad school"). I think also saying what kinds of courses you would be interested in at grad school, and what kinds of projects the group does you would be interested in, is valuable information (just look at their website or recent papers and say which of those is the most interesting to you) -- even though this is work you haven't done yet, you are showing you have some ideas and a direction and are not going in blind.

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