text:
I got a very bad review from one reviewer, but the paper was not rejected
Some time ago, I submitted my paper to a reputable journal (but far from being one of the best in my field) and have recently received two reviews. One review was positive overall and the reviewer recommened to accept my paper. On the other hand, the second reviewer was very critical of my work. The reviewer explained, in great details, why everything was wrong with my paper (even including the notation I use) and why such research is absolutely pointless and irrelevant. Moreover, the reviewer also criticized my other publications, implied that my research is way below the international level, and recommened to reject the paper. Despite such a bad review, the editor did not reject my paper and asked me to provide a revision without any additional comments.
To be entirely honest, I would've preferred my paper to be rejected after such a review, because I think that no revision can change the second reviewer's opinion about it. In addition, I do not really know how to revise my paper. I can change the notation and rewrite some parts of it in response to some of the comments, but I cannot change the topic of my research to make it relevant in the eyes of the second reviewer.
What would you do in my situation? I'm thinking about writing an e-mail to the editor in order to ask their opinion about the situation and what they would consider as a proper revision of my paper.

Did the reviewer who wrote the positive report suggest any changes they wanted you to make?

The first reviewer just asked to correct some typos. That's it.

"Moreover, the reviewer also criticized my other publications" At that point, the reviewer crossed a line -- that's clearly unprofessional. If it's a good editor, they would view the reviewer's recommendation with a grain of salt. They would check carefully whether the reviewer's claims are justified and you have addressed them properly. Maybe they will invite a third reviewer for the revised version, and ignore the negative reviewer if the others are positive.

"because I think that no revision can change the second reviewer's opinion about it." Your job is to concince the editor, not the reviewer. Often this is best achieved by addressing the reviewer in view of the editor. But sometimes the only way is to address the editor directly and convince them (not the reviewer) that you are right and the reviewer wrong.

I recently had an article in a top journal returned with such reviews.  One reviewer was positive, and the other seem to want to get rid of the paper by any means.  So he/she made up some rubbish.  I re-submitted my article and provided concrete evidences that counter all the rubbish comments made by the negative reviewer.  It is up to the editor to decide whether we are telling the truth.  Different to your situation, the editor knows I am a reputable author -- no, I don't use the name Santa Claus in my papers.

If any professor in your department is an editor for some journal, you should consult them in addition to us, because you can show them the reviews and your correspondence with the editor, whereas you should not show this material to the Internet.

Relevant line of thought, though not a duplicate.

My impression is that the editor didn't do their work properly here. Surely in such a situation if they ask you to revise, they should explain a bit what they make of the contrast between the two reviews. They may have been put off by something in the critical review and may not find it important that you address that one in detail, but then they may want to see a convincing rebuttal of that review. They should have indicated better what they expect you to do, and therefore to me it seems asking them for more detailed advice makes sense here.

Depending on how big your field/area is, the world might be small. It might be worth reflecting (after that paper) if the research is really relevant, or how it can be made more relevant.

Revise and resubmit just like you would any other manuscript. Address Reviewer 2 point-by-point as best you can. Responding to conflicting/unfair/blowhard reviews is part of the game.

Since

the editor did not reject my paper and asked me to provide a revision

you can revise the paper and resubmit with a fair chance that it will be accepted. Respond to all the comments in the favorable review, either by changing something or explaining in your cover letter why you didn't.
Try to put the  tone of the unfavorable review aside so that you can decide whether addressing  any of the comments there will in fact improve your paper.
You could write the editor in advance to ask if this is a reasonable strategy. You don't have to withdraw just because one reviewer was unprofessionally nasty.

"Try to put the tone of the unfavorable review aside so that you can decide whether addressing any of the comments there will in fact improve your paper." I find not uncommon that, in such case, when you take a step back and breath calmly, you realize that said reviewer has a point indeed. One that is, perhaps, poorly expressed, excessive, etc., but a valid point nonetheless.

Email the editor first.  The editor clearly did not auto-reject after the bad review, but you got a revise-and-resubmit.  These are common, and sometimes a revised paper is accepted and sometimes it isn't.  You are under no obligation to revise or resubmit, but it's best to seek out more clarity on the editor's decision.  It is also possible that the editor made an error, and you want to avoid wasting the effort of a resubmission if it will be rejected anyway.
You mention that reviewer 2 attacked your research quality and your previous contributions.  This is not acceptable behaviour of a reviewer.  The reviewer's job is to assess the work that is submitted, not to critique the scientific qualifications of an author.  Truthfully, if I were to be judged for my more recent papers based on past performance in conferences I'd never get anything accepted, and that simply doesn't make sense.  People learn and grow.
What you need to do is carefully review everything reviewer 2 said.  Sort the comments into groups: "I agree with this comment", "I do not agree with the comment, but that is because the reviewer misunderstood the paper", and "I do not agree with the comment, the reviewer is wrong".  For each of the three categories, figure out the next step:
"I agree with the comment" - Here the reviewer makes a valid point.  Addressing this with further work will definitely strengthen the paper.
"The reviewer misunderstood" - this is sometimes your fault, not theirs.  Assuming that the reviewer was acting in good faith, then a misunderstanding on their part implies that your paper wasn't clear enough.  Clarify it, since it will strengthen the paper.
"The reviewer is wrong" - Be careful but confident here.  Don't just assume that the reviewer is always wrong and that's that, but also make sure that you don't just agree because the reviewer said.  For these comments, you need to write a professional rebuttal in your resubmission letter: "We respectfully disagree with the reviewer.  Their claim X is in fact not a claim that we have made.  We have not modified the paper for this reason".

There is a fourth category "The comment is irrelevant" - which is where comments about previous work should go. If I was feeling bullish and confident, I'd say as much in the rebuttal, but probably here I'd just ignore the comments, or highlight them in the cover letter to the editor, rather than in the response to reviewers.

If you're feeling tricksy, there is another possible response to "the reviewer is wrong": reinterpret the reviewer's comment as a suggestion that's not wrong (maybe an improvement you wanted to make anyway), even if that slightly strains the natural and ordinary meaning of the words the reviewer used.

Meanwhile, I'm actually a bit uncomfortable with the interpretation of "'The reviewer misunderstood' - this is your fault, not theirs". This particular reviewer has pretty well proven engagement in bad faith. So they may be throwing chaff like this at the author just to make life difficult for them. Possibly expanding those points might make the paper actually bloated and worse.

It's not always your fault when a reviewer misunderstands something about the article. It has happened to me to receive a negative review in the lines of "no explanation about X was given", whereas there was a whole paragraph on the Methodology section dedicated to explaining exactly X. The same reviewer did this two or three times on the same review, which led me to conclude that he had not properly read the article. Of course this is not the case for the OP, but you can't say 100% of times it is the fault of the writer.

These are fair comments, I'll edit the answer

There's a good chance the editor is implicitly asking "what do you think of this review?". When the reviewer said everything is wrong with your paper, they want to know why you think the reviewer is wrong; when they criticize your notation, they want to know why you use the notation you do; when they say such research is pointless, they want to know why such research isn't pointless from your point of view.
So it's the same as with every revise decision: make the changes you think are justified, and rebut the comments you think are not justified (or provide reasons for why you are not making the changes). Remember that you are not trying to convince the second reviewer that your work is good - you are trying to convince the editor, who is able to overrule the reviewer if it comes to it. Your paper is not rejected, so there's still a chance it will be accepted, but you will need to provide a strong "response to reviewers" document.

Follow the advice and submit.  The editor feels your paper is worth including so just revise and submit taking into account the specific and constructive comments of the negative reviewer.  Not the general ones like - this stuff is worthless because the other reviewers and editor did not agree, and think what you have is important.

"The reviewer explained, in great details, why everything was wrong with my paper (even including the notation I use)"
It is up to you. The editors clearly think you should have a chance to respond to the reviewers, then they will evaluate if your paper is wowrthwhile for publication. The review is not a democratic process, the editor may at any point overrule the suggestion of the reviewers. The editor may even have an idea about the topic you are discussing.
Please divide the personal from the professional aspects of the review (unfortunately, you should not need to do that, but thank to one of the reviewer you have to). If you think it is worthwhile, address the technical/scientific points the reviewer raised (yes, even briefly the notation, for what it matters as long as it is consistent you can note gravity b instead of g). About the ad-personam attack, better get used to them: unfortunately, in the academia there is a lot of personal pride at stake. Think about how a third impartial person would see the discussion, if done live on a stage: one guy making a huge preamble, with many details on a techincal level, to conclude with a personal attack.
I see two possibilities:

If you respond with the same personal level, it will quickly go down
the road "how boring, cockfight between expected-to-be-smart people
discussing who has the bigger ego". It may be good for your ego, and even for your career, polarizing the line of thoughts;
Answering on the technical level only, the reviewer will have his ego satisfied, good for him, the general audience will think "interesting discussion, but that ego-driven professor seems an a***e".

Good luck, if someone spend so much time and emotion reviewing your paper it is somehow a good sign, formally it means your paper is questioning some scientific belief (held by someone) and the same someone had to spend time in proving them true.

To receive such conflicting input from the reviewers could indicate that either reviewer 1 (with the positive response) did not do a good job in reviewing and overlooked all of your submission's shortcomings, or that reviewer 2 is biased against you — or a combination of both. If reviewer 2 will also do the second round of reviewing (which is quite likely), I would assume that they would not be much more favorable if you did not change the whole article to a large extent. This could lead to more rounds of revisions, and a very long and time-consuming journey until your article might finally be published.
So if I were in the same situation, I would consider resubmitting the whole thing to a different journal (preferably with a double blind review process, thus preventing reviewers to judge your current submission based on former work (which is bad practice)). It might make the whole process less complicated and time consuming in the end, even though it may feel like admitting defeat towards reviewer 2.

Unfortunately, there are no journals with double blind review process in my field.

Occasionally in academia, one professor has a personal dislike of another researcher or a dislike of a research approach, and will tend to give negatively biased reviews to any such manuscripts. Such reviews are not an objective evaluation of manuscripts. It is the job of editors to consider the bias in such reviews when making decisions about individual manuscripts.

The reviewer explained, in great details, why everything was wrong with my paper (even including the notation I use) and why such research is absolutely pointless and irrelevant. Moreover, the reviewer also criticized my other publications, implied that my research is way below the international level, and recommened to reject the paper.

This criticism of other publications seems to indicate some sort of bias. It is entirely possible that the editor does not fully agree with this reviewer, and might choose to accept your paper if you provide an adequate response to the reviewer's criticisms.

If I were you, I would first email the editor to ask what happened to me, why there are two different responses, which may not make your article acceptable but it will make you feel comforted, then the editors will acknowledge their mistakes and correct.
Next, I will revise my article to the wishes of the second reviewer (to the best of my ability). Then, I will submit my article to other reputable journals. I think the results will be better.

which mistake are you talking about?

On the other hand, the second reviewer was very critical of my work. The reviewer explained, in great details, why everything was wrong with my paper (even including the notation I use) and why such research is absolutely pointless and irrelevant. Moreover, the reviewer also criticized my other publications, implied that my research is way below the international level, and recommened to reject the paper. Despite such a bad review, the editor did not reject my paper and asked me to provide a revision without any additional comments.

If the paper is not rejected while one of the referee reports is negative then the journal is not really "reputable". You do not gain much by publishing there. So I suggest that you withdraw the paper, improve it as much as possible and submit elsewhere.

Or it is differentiated and does not automatically reject but has a more nuanced opinion; e.g. seeing that the second review seem strongly biased

In that case the second review is not sent to the author.

Why not? Hiding reviews from authors seems rather live a shady way of doing things, not what I would expect from a reputable journal

@Mayou36: Judging from what is written in the OP the report is clearly unprofessional and the choice of the reviewer was wrong. Such reports are not sent to the authors unless the editor wants to insult the author.

On the contrary, I know about many examples where the most reputable journals in my field (physics) have sent out a pair of reviewers that are exactly like OP had described.

@Ezze: Where exactly did they send the reviewers?


text:
How should I talk to professor if I have been absent from project for weeks due to mental health issues?
I am an undergraduate student. Four weeks earlier my professor asked me to work with him on a literature survey project. He asked me to update him every week about the status of the project. The project I am talking about basically marks the gender gap in an interdisciplinary field and I have to go through research papers and separate males and females, contact people rather than doing any research work. I worked on it for the first week showed him the excel sheet I made, but he didn't express any thoughts on that rather told me few other sources where I can get more data and answered the question I asked. I knew I was not very motivated to work since all I have to do was copy-paste data from sites to excel.
Then, my family members got sick(with most symptoms of COVID), I have to involve myself in that and it really brought my motivation down to zero. I couldn't bring myself up to manage time, I suffered from anxiety and depression in the second week, this disrupted my thinking I guess as I didn't inform my professor about the project. And in the third week as well I made no contacts with him and no progress in project.
But now when I think I am a little better, I see this as being irresponsible from point of professor and that unknowingly I have done a harm to my reputation since he is one of the best professors in my department and he has always been supportive to students.
I still haven't prepared enough data, what should I do now, how should I talk this out to my professor? Would an apology be enough? Should I ask him if I am still allowed to work on that project?

Good luck to you. Remember your advisor is human too.

@A rural reader Thank you sir, but I don't quite understand.

Related: https://academia.stackexchange.com/q/158913/68109

Are you being paid as an undergraduate research assistant or is this something you volunteered to do?

@kjacks21 I won't get paid, my professor asked me for the project to which I agreed.

Not. Do not mention the exact cause anyway.

what should I do now?

You should contact your professor and let them know what happened.

How should I talk this out to my professor? Would an apology be enough? Should I ask him if I am still allowed to work on that project?

You should just tell them what happened, why you were absent, and that you would like to continue working on the project. While copying data to Excel is nice, it's certainly not more important than your family! Your priorities were in the right place, and what happened is not your fault. You should perhaps apologize for not letting them know earlier what was going on, but certainly not for anything else.
It's annoying when students ghost you, but in your case, the situation absolutely warranted it. If your professor is a reasonable person then they would understand and be nice about it. They may not let you continue working on the project because they already found someone else that would help, but it's definitely worth asking them to continue if you're interested (though it sounds like you're not that interested in this project).

I agree that their priorities were in the right place for everything except the lack of communication.  You don't have to go into great detail - just a quick email explaining that you're having some personal challenges that will be impacting your performance.  If someone just disappeared from their job for a couple of weeks, you wouldn't expect them to be able to step back into it as if nothing had happened and you wouldn't be surprised if their employer was unhappy about the situation.

@Spark  I guess I don't have any other option, I will do that. But say he said that I can't continue the project, would it be so bad for my research journey?

It's definitely not a positive outcome - I wouldn't expect to get a glowing reference from this professor, for example. However, life often happens when we least expect it, more so in the past year than in other times. Get up, dust yourself off, and continue looking for research opportunities - that's all you can do really!

@Spark about the last line, Say if I am not very interested in the project, shall I only send an apology email? it will create an even more negative impression.

It's all about positioning. You can say something like"at this point in time, I do not feel that I have the capacity to continue working on our project. I understand that this is a major inconvenience, and apologize for not contacting you earlier."

Update your professor about why progress has not been made and what your next steps are. If you need more time to focus on your mental health, then let your professor know.
One other important thing we should make clear: you are doing this work on your own time, as opposed to being paid. Research mentors or professors with unpaid research assistants should be grateful for any time you can devote to the project assuming you do not take their time for granted. It appears thus far that your professor has not invested a lot of time into the project with you, so you are fine. We should also note that you have no contractual obligation to do the work.

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How should I prepare my write-up for an internship program?
I am going to apply for an internship programme. It is written that the selection is based on write-up. To apply for the program I need to include an write-up with 1200 words. Also, some outlines of the project is given in the website.
But I am little bit confused. Which things I should include in the write-up? Should I include only "Why I am interested for the program?" or the part which includes what I have read earlier about the topic that I  want to pursue in the internship?
I have no idea.  Please advise me.
Thanking in advance.

You could Include the following (in order) -

A very brief introduction about yourself and the motivation behind working in the field of the internship you are applying for.
Where you have interned/Projects done
The skills developed from point 2.
What you hope to acquire from the internship and how it will benefit you (very short sentences)
An ending touch.

I did not participate an intern before. So, can I replace 2) with self learning? What does an ending touch mean?

Yes, you can definitely replace that with self learning. Probably mention a book or paper you read? An ending touch meaning just a sum up of why you need this role and what makes you a great fit for it!

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text:
Signed up for a talk, not enough material
I am a PhD student in pure mathematics about 4 months into my program. After consulting with my supervisor and some peers, I have signed up to give a 20 minute talk at an online conference, which has invited final year undergraduates writing a bachelor's thesis, masters students, PhD students and postdocs to sign up and optionally give a talk.
My supervisor and peers have told me that it is a good idea to sign up as it's really good for my CV. At this point in my program however, I am mainly learning about my field, and haven't produced very much original research. I have however come up an (in my opinion) interesting question that would make for a good talk if I had a partial answer, but at this stage I have not had much time to explore the answer.
The conference is in about a month's time, and I think it would be a great exercise to write a talk giving an introduction to the field and how it came about, but at this stage this is all I could do, and even this would be quite a task to do in a month!
The sign up page for the conference does encourage works in progress however, but I am not sure how much I can do in a month.
My questions are:

Was it a mistake to sign up for this talk?

If, in a week's time say, I feel that I can't present any partial progress, can I ask to be removed from the schedule, or is this unacceptable?

Is there anything about the nature of this conference that suggests new research results are expected? I've seen many talks that were basically surveys of a problem and what others (and often, but not always, the presenter) have done toward solving the problem. In fact, my Ph.D. advisor gave such a talk (I didn't hear this particular talk, however).

Did you ask your supervisor these questions?

I would say it's a bit unusual to give a survey talk instead of a research talk if survey talks weren't specifically solicited, but it happens sometimes, and as long you can do a decent job, it should be a good experience and opportunity to introduce yourself to the community.

In twenty minutes, presenting a new interesting question without answers (but maybe with some cool examples) is perfectly fine for a talk in pure math.

Twenty minutes is not long for a talk.  Moreover, most students drastically overestimate how much material you can put in a short talk.  Giving a short introduction to the field and how it came about will be more than adequate to fill that amount of time, and would be a perfectly reasonable topic for an incoming PhD student with only four months in the program.  Time permitting, it might also be nice to mention your research question and the general avenues of inquiry you are going to use to solve it.  You should also leave a few minutes for questions and comments, particularly since this will give researchers in the audience an opportunity to offer suggestions on your research idea.
So no --- you didn't make a mistake signing up for this talk.  Don't bother trying to cram out new research results for it --- just stick with your plan to give a short introduction to your field and a brief outline of your research question.  That should be enough to hold people's interest and solicit some questions, and the twenty minutes will go by in a flash.  I doubt a short talk like this is really going to help your CV all that much, but it will be good practice for giving conference talks, and if you are lucky, it might precipitate a longer conversation on your research idea with one or more of the researchers in the audience.
(And as Buffy points out in the comments below, no one will be upset with you if you don't use all your time --- running over time is bad, but having time to spare is fine.)

And, no one will hate on you if you don't use all the time. Running over is a different story, of course. Good advice here.

@Buffy comment only? seems humble of you. there's a flag to convert answer to comment. i want an anti-flag: to convert comment to an answer!

@BCLC, no, just an addendum to writer Ben's good advice. But thanks for the thought.

@BCLC: Comment folded into answer --- consider it anti-flagged!

Ben and @Buffy : nice

@Buffy While I agree with this, you probably don't want to go drastically under time. If your 20 minute talk is 15 minutes, most people probably won't even notice. If it's 5 minutes, that might not reflect well on the presenter.

Twenty minutes is not long for a talk. especially for people who are not used to talking on conferences, or teaching etc. Stressed out, they prepare material for 6 hours and after 20 minutes they went through the first slide. OP should plan for 12 minutes and be done.

For the past 30+ years, I've been giving talks at conferences, both academic and non-academic. Many, many times by the time I start writing the talk, I start to think "how am I ever going to fill the allotted time". Then, when I have my first draft ready, it's more "OMG, I have 60 slides, and only 20 minutes".
20 minutes is not a lot. If that includes time for questions, and an introduction, time will fly.

Was it a mistake to sign up for this talk?

That seems unlikely. Unless you show up without any slides and no preparation because you planned to write the slides the night before, but got drunk in a bar instead (I've seen that happen).

If, in a week's time say, I feel that I can't present any partial progress, can I ask to be removed from the schedule, or is this unacceptable?

It happens frequently that people ask to be removed from the schedule, for many different reasons. For conference organizers, this is part of the game. I was in charge of the program of a conference last year, and I had to make last minute changes because a few speakers cancelled. Conference organizers cannot force people to show up and speak. But you would do yourself a disservice.
Now, I understand you're nervous. I still am before giving a talk -- that will never go away. But whatever you're talking about, you are the expert. You know more about it than the audience. Presenting at conferences are a good way for people to get to know (of) you.

I would like to highlight one fragment: “I still am before giving a talk -- that will never go away.” — This is true for some people, but others learn to not be nervous at all over time.

Was it a mistake to sign up for this talk?

No, you will be soon in exactly the same situation you describe, but the talk will be in front of the heavy weight of your topic, not BSc, PhDs and the likes.
Why do I say so? because a talk will and must be showing a work in progress, at different stages of completion. If the work is done, then you are presenting a paper you completed, which is relatively "boring". I can read paper, I prefer to read a paper rather than attending a talk with an introduction by the author to the paper (off-topic: same reasoning applies to the fact I do not like most of the travel guides, where you are suggested with which mood you should enjoy a certain view/building/landmark).

If, in a week's time say, I feel that I can't present any partial progress, can I ask to be removed from the schedule, or is this unacceptable?

You are facing a big climb: preparing your first presentation. In the next years you will use and reuse this material, as the core building block of your talk. Or at least, you think so. In reality, you will discover that you will be basically rewriting everything and even having only marginally acceptable partial progress, this will translate in a humongous amount of work in the last week. This first time experience, however, will help you survive for the future iterations of this process.
Jump in the unknown. Present your problem/goal (2 minutes). Present the literature you are based on (3-4 papers, 5 minutes). Present literature open points (3 minutes) and your questions (3 minutes). Present your approach (2 minutes), without any partial progress (0 minutes). Present your partial progress (2 minutes, if any).
One slide  = one minute (or more).
Good luck!

ps: people will be quite annoyed if you will take more than 20 minutes, but they will be happy if you leave them 5 minutes or so to interact with you with questions and remarks. This apply also to the big conferences: leave 15-20% of the total allocated time for questions ...

In my years of giving and listening to talks, I've never heard someone complain about a talk that was too short.  If you have ten minutes of material for a twenty minute talk, that's perfectly fine; there will be a few minutes of introduction, then your ten minutes of talk, then a bit of time for questions perhaps, then everyone will have a few extra minutes to get to their next talk - it will be refreshing for folks to not have to rush to the next talk after a 28 minute talk, as often happens.
And of course, don't be surprised if your talk does go longer than you expect.  They usually do!

I will try to provide general answer that might help you later too.
You have to ask yourself: what is the function of your talk and who is your audience?

I have signed up to give a 20 minute talk at an online conference, which has invited final year undergraduates writing a bachelor's thesis, masters students, PhD students and postdocs

This suggests to me very general audience, and function of this event is more education than presentation of research results. You can choose other function/audience but you risk to have a mismatch.
Assuming these function and audience, 20 minutes is plenty to (for example):

educate your audience on importance of your field, and how it connects to other famous fields
present "an example of research-level question and process to answer it"
educate undergrads/other PhD students on your process. Undergrads will want to learn how math PhD looks like in practice

As soon as you understand the function of your talk, it shouldn't be hard to see that you will provide value/be useful if you just teach people about your field and process. It is still hard though and require polish (short talks especially important to polish).
PS: still you can decide that the function of your talk is to present Fields medal-worthy work, and aim to fulfill that function. But should you?

Do you have any MSc project work or the like? I pretty much had the same experience and chose to talk about the project I had done before starting the PhD (at that point this could have been a basis for the PhD but ultimately this turned out differently). It wasn't super original and I felt slightly bad about it when preparing, but it turned out to be a good experience.

Those types of conferences (including students and PhD candidates) are often designed specifically, or at least partially, for young scholars to test their findings on training grounds.
This is more than likely that you do not have enough material, or that it's not original, but that's okay--this is a great opportunity for you to talk about your problem/field of study:

By preparing a talk you learn how to rethink and summarize the issues you tackle, which has a potential of deepening your understanding.
Feedback from your peers and superiors will help you improve your research--by pointing out flaws, suggesting new venues for exploration etc.
Chances are that to many among the audience those issues are new and they can learn something--this is inevitable in all domains too broad to be mastered by one person, and math is indubitably one of those.

There is a great value in being able to define and conceptualize problems concisely and simply for a variety of reasons, and this is an opportunity for you to learn how to do so (incrementally, of course).


text:
I've developed a close relationship with my past graduate teaching assistant, and I'm very interested in her: is it appropriate to pursue this?
The graduate teaching assistant I'm interested in taught one of my introductory courses. She needed research assistants, so I decided to join her lab. From all this we became very close. Ever since the end of that semester, we talk nonstop.  It's been more than a year since I was in her class, but I am still an undergraduate. I am also applying to essentially the same graduate program she is in. We are not working with each other right now, but it is extremely likely we will work together again.
Would it be inappropriate to pursue this? The people in her lab know me. Would this make her look bad?

Since you are not working under her supervision right now, then assuming you are both adults, you are in the clear ethically, legally, and university policy-wise.
The only real caveats are that if you end up involved in a romantic relationship with her, she will not be able to write you a letter of recommendation, and will not be able to assume a supervisory role over you in the future. However, academia has plenty of dating and married couples working in the same department and sometimes on the same project. There are well-established norms for dealing with such situations, and no one thinks this is anything to get worked up about.
Good luck!
Edit: as I somewhat feared would happen, people are volunteering unsolicited advice about whether it’s a smart idea to enter a relationship with someone you may end up sharing a workplace with in some hypothetical future, based on further hypotheticals about what would happen if you end up splitting up or whatnot. Since I respect you as a fully autonomous adult capable of making your own decisions, I have restricted my answer to things you actually asked about and that are in the scope of the sort of advice this site is designed to offer. I would advise others to do the same. There’s nothing inappropriate about a romantic relationship in your situation. It may be a good idea or a bad one, wise or unwise based on numerous factors as with any question about whether two people are a good match for each other. But that’s not what you asked about, and has no connection to academia.

Comments are not for extended discussion; this conversation has been moved to chat.

Would it be inappropriate to pursue this?

No. It is appropriate IMHO. It is also reasonable that we develop feelings and form personal relationships with people which life brings us into prolonged contact with.
If she has some official capacity in the lab, and you two do hit it off, you might need to mention the situation to the lab's ranking researchers; and that's still just fine.
But - don't use your lab / professional spaces as a stage for propositioning her about a relationship. Ask her to have chat in private, or somewhere out in the open etc., to talk about how you feel (or to ask her out on a date etc.)
Also, bear in mind she might only interested in you as a friend and a colleague, not as a romantic partner. And then it will be you who may feel somewhat awkward.

The people in her lab know me. Would this make her look bad?

Being in a relationship with you - no. But being the focus of romantic attention while doing her job might be embarrassing for her (or for others), hence my advice above.

Pro tip: don't start out with "I have all these feelings for you and x and y and z". Aim very very small: "Hey, would be interested in going out on a date? No? Cool, catch you on Monday".

@SteveBennett: See edit.

Also, bear in mind she might only interested in you as a friend and a colleague, not as a romantic partner. And then it will be you who may feel somewhat awkward. Workplaces have always been venues for people to meet and form romantic attachments - if everyone worried about feeling awkward, people would be routinely very lonely. And very rare.

We are not working with each other right now, but it is extremely likely we will work together again.

Really, this isn't the problem with the scenario, the problem is if she is currently in a position of power of you. Actually, you'll be much better off when you graduate. As a comment points out, this might mean you don't work in the lab anymore. If you don't work in the lab at all, nothing applies and you're both adults.
It sounds like the TAship is no longer a problem, either.
You haven't tagged a country, but most places don't outright ban grad student-undergrad relationships, the rules exist to protect you in this scenario.
If you enter a formal relationship, make sure you (she) tells her PI so everything is aboveboard, even if you don't work with her, just in the the same lab.

Would this make her look bad?

Bad enough for her to get in trouble or shunned? No.  It's possible she might encounter some light (inoffensive) teasing.

"Is it somewhat looked down on for a grad student to date an undergrad? A little, I won't lie to you." - how does that actually work? Do (most) people only date students at the very same stage of education as themselves? Otherwise, one partner finishing their undergrad studies before the other does must be an utterly common occurrence that virtually every long-term student couple goes through.

@O.R.Mapper I don't know about you, but the average age to start my grad program is probably around 24 or 25. That means that the gap between a random grad student and a random undergrad could be big enough to be teased out in the real world. Anyway I don't want to spend too much ink on such a minor issue.

@AzorAhai-him-: Maybe I don't quite understand what is meant by "teased out" here, but at the age of 24 or 25, I don't see anything unusual with a difference of +/- 2 years. But even so, why assume a gap at all? It even seems totally normal to me that two students are at the same age, but one becomes a grad student a year earlier than the other.

@O.R.Mapper Oops, I meant "teased about."

Re: your last paragraph: I encourage you not to underestimate the hostility and devaluing that women can experience for doing the exact same things that might result in men encountering light teasing.

@GregMartin Fair enough.

@O.R.Mapper I would say it's more about when/how the relationship was established. If you start as peers then it's usually nbd if one person is ahead a bit. (On the other hand, a lot of couples also break up when one person graduates anyway, or shortly thereafter, since a lot of people move. The two body problem is alive and well.)

Here's three things you should spend some time thinking about:

Does your university or your lab have any relevant policies? In this case there's probably no relevant policy, but if there is that could immediately answer your question.
Does your current relationship have a strong mentor/mentee flavor?  Would you say that you look up to her?  Is she someone you turn to for professional advice?  This kind of unequal dynamic can be really dangerous for forming a healthy romantic relationship.  Furthermore, you may end up regretting losing a mentor.
Is pursuing her romantically likely to be awkward for her or make her feel uncomfortable in her workplace?  Do you have any reason to think she's interested romantically in you and not just interested in serving as a professional mentor and friend?  Can you bring this up outside the workplace in a low-stakes way where it's easy for her to turn you down?  Are you going to be able to immediately drop it and not make things weird?  Making a pass at someone at their workplace is a big no-no, whether they're a colleague or service workers.  People deserve to feel safe and comfortable at their workplace.

Your second and third points, although perfectly valid, don't seem related to the actual question or the subject of this site. I mean, it's decent advice to give a friend, but doesn't seem relevant to whether there is any legal or ethical issue involving the university.

I think they do speak to ethics: it's unethical to date your mentees and it's unethical to create a hostile work environment, and questions 2 and 3 get at whether either of those things might happen (or might not).  I'll admit that there's probably not much special to academia here, similar issues would come up with coworkers in other jobs (though undergraduate students are a bit of an unusual case that don't have a great analogue at other workplaces).  I think this question would be on-topic at academia or workplace.

I recommend that you keep your professional and romantic relationships separate.  Even if it is not ethically required, it is a good way to keep your work from taking over your personal life (or the opposite).
It seems that, as there is currently no actual working relationship, you have no ethical problems with starting a romantic relationship.  But expect that you might need to choose between the two later, either for practical, ethical, or policy reasons.  Read your university's policies.

Agree. And might not do any harm for you to frankly discuss the looming situation with the TA too. The problem isn't just university policy - it's the age-old problem of the works relationship, be it marital, romantic, blood-relation or just friendship. Others in the work group will not tolerate third-party tackling when they make a fair challenge on one of the two people who are close. Put yourself in their shoes. Every time you challenge A fairly on a work-related thing, B comes running around answering for them or giving you a bad vibe.

Generally, dating inside the same institution is fraught with problems, especially if between levels. It always was that way, do not subscribe to the illusion that somehow things were better once, just because there were fewer regulations (which were introduced for a reason).
It's certainly not impossible, and there are good examples for that, but there are all kinds of problems that can emerge and it carries a clearly elevated risk.
Even if the institutional code is lax about that (which is unlikely, they become more stringent lately in view of changing societal norms), when something goes wrong and emotions go haywire, it can maximally escalate, to the detriment of both parties. In earlier times, the risk used to be concentrated on the more junior party, today, the more senior person is equally exposed.
Both of you, before you partner up, should be acutely aware of this. Especially if you both apply for the same grad program, there is significant potential that things could go wrong (competition, different rate of progress, authorship disputes, etc.). It might be good to stay away from working together until it's clear that you really get along even in times of crisis.
Proceed at your own (and your potential partner's) peril.
With sensitive issues comes the need for careful handling.

This is so sweet!
Before you proceed, do you know anything about her status, like maybe she has a significant somebody or obligations that would hinder the process?
Other than that, don't fantasise too much about the outcome, keep it lite.
Wait for her to be alone or make an appointment and invite her to go for a coffee. And then, just be a gentleman, don't jump ahead.
Work situations are not ideal and you may feel socialy inept, don't worry.
Give it a go and handle it with respect.

I had assumed that since the two of them talk a lot, the OP already knows sufficiently enough about her personal life to know whether she is involved with someone or not. Therefore I presume she is not. I wish OP the best and hope it works out.

As a general rule, it doesn't often work well when adults enter into a relationship and one has power over the other. In this case, she has power over you. If I were you I would wait to act until the application process is over. Once you are also a graduate student (if you are accepted into the same program in the same place), then you may or may not want to pursue a relationship with her. But there is always the chance that your attraction to her is in part because of her power over you. The only way to know is to wait until you have more equal status and see how you feel.
One respondent above asked what country you are in. A different way to phrase this might be to ask what cultures each of you grew up in. This is important, as power dynamics in relationships are different in different cultures.
You also have not made clear what your gender is (presumably because you are male). All of the other respondents seem to assume that you are male (presumably because they are male). So if you are not male, you may want to specify that for us. If you are male, do not assume that dating you—or even being asked on a date by you—will be an uncomplicated situation for her. Most female graduate students in the sciences have to negotiate a fair amount of misogyny and male privilege and she might see your advances as yet another inconvenience in a professional field full of inconveniences. Dating you might also make her look unprofessional in the eyes of her peers. If you do care about her, you will work hard to understand these things, and choose your actions with not only you and your feelings but also her and her career in mind.

I'm 20, so I might be too young to answer this. But yeah, there are few things you might want to consider:

Is it really a deep feeling, or is it just attraction?
Does it matter to you and her, what others think of you two?
How will this impact your future, and her future?
Most importantly, what's her take on this?

If you know the answers to these, then I guess other factors such as peer thoughts, gender, age gaps, country, culture etc. disappear, since all of them can be tackles in some way or the other if both of you are looking forward to it.
Hope this helps, and Good Luck :)

I'm a guy, and speaking for myself I can imagine heaven much more easily than hell, and that is a problem because I choose what I imagine to be the best option, but my "best" is biased by my dangerously positive brain.  By that I mean that I can easily envision all the ways I might be happy, but it takes intentionality to consider those futures that could turn my life into a living dumpster fire.
I find it useful to ask "what are the ways this can turn into a dumpster fire".
These are some (non-exhaustive) examples of my consideration of "dumpster fire" outcomes:

You approach it like a (dumb) guy and she reports it as hostile work environment, so you are ejected from the program, get a reputation that is public enough to destroy your career prospects, and you have to flip burgers for the rest of your life.

or

It works well for a bit, you get romantic first, but then you act like a (dumb) guy and she reports it as sexual harassment, so you are ejected from the program, get a reputation that is public enough to destroy your career prospects, and you have to flip burgers for the rest of your life AND she can pursue suing both you and the university in civil court.

or

It works well for a bit, you get romantic, and she gets pregnant, and then you act like a (dumb) guy and she reports it as sexual harassment, so you are ejected from the program, get a reputation that is public enough to destroy your career prospects, and you have to flip burgers for the rest of your life AND she can pursue suing both you and the university in civil court AND you are paying child support for the rest of your life and bring that into any future relationship.  Or she has to drop out of the program.

This is just a two-body problem analysis.  You can think through more two-body dumpster fires.
Let's add a third body, which could be any one of:

Her advisor
Her rival
A rival to her advisor who wants to attack the advisor using you
and/or her as a weapon
Another student at your level who also has interest

How can they put gasoline on any of the above?

Someone could falsely report your relationship as professional conflict of interest where you get an unfair advantage.  or where Where she gets an unfair advantage.

How can a lawyer turn this from a dumpster fire into a flaming sewage hell-storm?

If you have met and talked with any reasonably smart lawyer-ish people, this question itself should scare the holy carp out of you... they can destroy you forever in ways you can't possibly begin to imagine.

If, and only if, you can navigate a few of these "potential hells" then you can also insert your presumed "heaven" with the possible outcomes.
UPDATE: So how do you mitigate it?
There is plenty of advice in stuff up there that seems judgy or paranoid but actually engages some of these forms of risk.

To make sure it is not a favoritism/pressure make sure there is no
organizational power that can be contrived in an argument (or court)
to show one is in executive authority over the other.
To keep it from being a hostile work environment, take baby steps,
and work on being aware of both how she reacts and how other folks in
the lab react.
To keep from being sued, be extra specially careful not to go
anywhere near anything romantic until there is some strong and
consistent-over-time trend toward acceptance and enough concrete
evidence (not hearsay) to let you defend yourself in a court of law
or even worse a brief university hearing in front of political sharks
who would gut you for an ounce of push toward their next advancement.
If it does get intimate, make sure you have both talked about and
agreed upon the method of "protection" ahead of time, so that
pregnancy does not become a disruption.
Show enough consistent (and possibly expensive in terms of time or
work) pursuit so that it is both widely obvious and any contenders
can elect to join battle or forfeit in order to reduce human-level
contenders.
At some point, chat with a lawyer about liability and such.  There
might be free lawyers on campus that need volunteer work and could
help you make sure you are not setting yourself on fire.

“Sonny, true love is the greatest thing in the world.” — Miracle Max, The princess bride

Sounds a bit paranoid, to be honest. But also an amusing list of worst case scenarios.

@henning--reinstateMonica - thanks for the edit.  I'm much more software, much less human communication.


text:
Which domain should I choose for my personal website?
I have decided to create an academic personal website, but I am unsure whether I should choose a .com or a .org domain.
I have read that .com is mainly used by commercial organizations, whereas .org is intended for nonprofit use. Hence from this viewpoint I should go for .org, as my website will be nonprofit. On the other hand, many suggest to opt for .com anyway (see e.g. here),
To complicate things further, I have verified that the domain namesurname.org is available, whereas namesurname.com is not.

Related: https://webmasters.stackexchange.com/q/2232/

Apparently, .com conveys a sense of credibility so that it would be the best option, regardless of whether you pursue things commercially or not.
It seems to me that .org doesn't suit a private homepage.
But for a private homepage, you can generally be creative and look for any other options that suit your biography (e.g. citizenship) or name (e.g. if your surname is Li, why not choose .li?) or interest (e.g. if you do research on marine life, then you could use .fish).
And for research-related purposes, there are also top-level domains such as:

.science
.engineering
.academy

Or .me, since the website is about you.

With ccTLDs like .li and .me, one should also take into consideration who controls it and how restrictions on who can own domain names under that TLD may change. (e.g. losing .eu domain names due to Brexit: https://www.gov.uk/guidance/registering-and-renewing-eu-domain-names-in-the-uk)


text:
Will getting the citation format minorly incorrect cost me?
I recently submitted a graded essay with the Harvard citation format and did not know that for single authored papers, the in-text citation follows as Last Name (Year) when starting a sentence or using it within a sentence. I accidentally used the full name instead for a few of my citations but have referenced every single citation in my bibliography.
Will this minor error cost me a lot? Is it plagiarism if I have gotten this wrong?

Plagiarism is when you take something written by someone else without attribution. That has certainly not happened here.
My best guess for what will happen is something between "literally nothing" and "almost nothing," depending on whether perfect referencing was a specific outcome to be assessed on the essay. It is, at worst, an incredibly minor, forgivable, understandable mistake. Don't worry about it.

At worst, an editor of a publication would correct it or ask you to. No one can predict what a professor would do and someone very picky about formalities might mark you down a bit.  But you could also be proactive and tell the professor about the error, offering to correct it.

Hi, thank you for your answer! I could definitely do that as well, and possibly should, but since there was a deadline for the essay and its been overshot, I don't think I'll be able to do so since the grading system is anonymous and the essay was submitted via Moodle.


text:
Promotion But No Change in Salary?
Recently I received notification that my application for tenure and promotion to associate professor was approved. However, I have not yet received a letter indicating what this actually means as far as any potential changes in salary. Is it unusual not to find out what those changes would be when you are promoted from Assistant to Associate Professor? Or do you typically not find out until the beginning of the next semester?

Congratulations! Typically the promotion has an effective date (say, July 1st), which is also the effective date of the salary change. Why don’t you ask HR or the admin staff at your department for the details?

@DanRomik, that's probably better as an answer. Especially the last sentence.

As stated above, your admin can tell you more than we can. But for us to even try answering, we would need to know at least where in the world you are.

For my own experience from a few years back: I was notified of my promotion in late April, informed of a future corresponding increase in pay in mid-May, and given the raise at the beginning of the following Fall semester. Your mileage may vary greatly based on institution type, country, etc, but I can attest that it is not necessarily unusual for your letter about promotion to be devoid of pay details.

My university's Board of Regents usually approves promotions in May but waits until July or August to approve the university's budget for the coming academic year. So a newly promoted faculty member might get a letter today announcing the promotion but might not find out about next year's salary until August.

in my country, salary for university staff is regulated nationwide, depending on your position and the number of years you have already worked at the university and you could look up your salary in a table listing.

It will depend on a specific department, I guess the best way to find out is to ask (the department head, or the business office of the department). For example, at our department, Assistant to Associate comes with "automatic" $5k salary raise, while Associate to Full has no automatic raise whatsoever...


text:
Can a Mathematics Undergrad do a Ph.D. in Bioinformatics?
I have a particular background, I studied biology and Chemistry in Highschool, some engineering in college, and I am now majoring in mathematics with a minor in computer science.
To some extent I feel like being able to do research is on my life goals bucket list. Diving into unexplored territory definitely excites me. So I somewhat feel I need to take advantage of the broad spectrum of fields I have knowledge of, although almost insignificant, it is still there. I thought maybe bioinformatics mixed with some data science algorithm research would be a good path to pursue. Would my prior experience in Biology give me an advantage? For example the ability to pick up technical biology concepts quicker?
Any insight and guidance on the topic would be appreciated.
Kind regards.

I also studied biology and chemistry in high school (and not thereafter), and in my opinion what I learned there is so little, compared with what biology or chemistry students learn at the university, that I would not consider it an advantage for moving to a field related to biology or chemistry. On the other hand, my background in mathematics might well give me an advantage in such a field.

Doesn't everyone study biology and chemistry at school?

Yes, let's just say I studied advanced Biology. For example, I know precisely what oxygen is for. I doubt everyone with a High School degree and college unrelated to biology does too.

Is this a question about US study?

@astronat This varies fairly wildly by location. It's quite possible (and indeed quite common) for someone with a maths degree in the UK to have studied no biology or chemistry at all since they were 16 years old.

@user3482749 yes, I know -- I am one of those people! But isn't a GCSE roughly equivalent to a US high school class?

There are a range of different things covered by the broad terms "bioinformatics" and "computational biology".
Some bioinformaticians are (like myself) biologists who use a computer to carry out their experiments rather than a pipette.
Some bioinformaticians are genuinely informaticians - they are interested in how information is stored, organized and structured. They are into databases, data formats, the right types of metadata that need to be collected etc.
Some bioinformaticians are basically applied statisticians. Then it is all about the stats knowledge.
Finally some bioinformaticians develop and build algorithms and new methods to analyse data in new ways, or automate or performance optimise previously designed analyses. These people far more likely to be computer scientists.
This is before we even consider mathematical biologists and biological theorists.

None of these paths is closed to you. I am of the "biologist with a keyboard" breed of bioinformatician, and have taken on students from a CS background in the past, and would taken on students from the CS/maths background in the future. But you will have work to do to learn the biology, and you won't get very far in my lab without developing an expert level knowledge of at least the biological area your thesis is in. UK PhD students don't as a rule take any classes. But I have send CS background students to undergrad classes in the past.
On the other hand, you are probably perfectly placed for a PhD in an algorithm development and optimization lab. You'd need to learn some biology, but probably wouldn't find it so onerous.
However, in neither case do I think advanced level high-school biology would make much difference. Our biology undergrad requires chemistry, and recommends maths, but does not require biology at high-school

A final word of warning - biology still very much has a culture that the reason to do a PhD is to spend your life in that field. Not necessarily stay in academia (although some of the older faculty might still be reluctant to take on someone who doesn't want an academic career), but at least start with a desire to stay in some sort of bioinformatics. I'm not saying its a good thing, but it is a thing.

Thank you for the insight.

Sure. There are many people who transitioned to Bioinformatics from a math background. As one example, there is a lot of machine learning being applied to various aspects of biology, and that could be a good fit for you.
But I would just caution that when choosing a Ph.D. topic, you should really think about what you want to do in the future, rather than what you've done in high school, or even taken a couple classes on in college. In a Ph.D. you'll anyway have to learn a lot about some very narrow topics and also (possibly) take some broad classes, so the fact that you know a bit extra about some biological concepts is not that important; a student without that background could probably catch up by reading a couple of books or taking a Coursera class, and both of you will be way behind an actual Biochemistry major (but still OK in the end). Pick your topic and advisor based on what sort of problems you want to solve for the next five years, and where you really see yourself ten years from now.

Caution: whether it's true that 'In a Ph.D. you'll anyway have to... take some broad classes' is country-dependent.

Dear @Jen, how important do you think choosing the "right" Ph.D. is? Most people don't end up working with anything related to their field anyways? So what difference does it make to work towards a Ph.D. in Bioinformatics, Data Science, Computational Science? All I want to do research publish a few papers, and get into the industry as a software engineer/developer.

@Dedekind This is not true for Bioinformatics. A far higher proportion of Bioinformaticians will stay in bioinformatics than CS grads stay in CS research. Sure, most people don't end up in academia long term (although its higher than many other subjects because to the shortage of qualified bioinformaticians), but its not a normal route to software engineer/developer. Data science is more common destination.

@DanielHatton True enough, thanks! Added a slight edit to my response.

@Dedekind - I guess I'm kind of confused about what you want, then. If you just want to work in industry as a SWE, you don't need a Ph.D., maybe not even a Master's (source: I was a senior SWE at a FAANG). If you want to do research, presumably you have some idea of what sort of research you'd like to spend 3-5+ years of your life doing, so my point was just that this is much more important than your coursework. Having said that, bioinformatics is pretty awesome, and I don't at all mean to deter you.

Would my prior experience in Biology give me an advantage? For example the ability to pick up technical biology concepts quicker?

No, not relative to most applicants for a PhD program in bioinformatics. There will be others who studied a quantitative field more related to biology than your math/CS pairing. Modern approaches to every area of biology involve increasingly quantitative methods. Graduates with undergraduate degrees in molecular biology, neuroscience, ecology, biochemistry, etc, are all potential candidates for bioinformatics PhD research, and they've all had an entire undergraduate education in biology beyond highschool. There will be students who initially pursued a medical/health profession and have taken courses in physiology. There will also be CS and stats undergrads who worked on biological or medical problems in their undergraduate research, people who primarily studied biology but with a minor in statistics/data science/bioinformatics, etc. High school biology, even "advanced" high school biology, will have you below the par for biology expertise in a bioinformatics field.
However, that doesn't mean it isn't possible for you to pursue bioinformatics (which itself is an incredibly incredibly broad area, the term itself has completely different meanings to different people). Instead, I think you'd be better off thinking about the ways that your prior experience in mathematics and CS gives you an advantage in bioinformatics, rather than a topic you last touched in high school. You probably have a lot to catch up on in biology, and that's just fine, as long as you realize how much there is to catch up on. We get quite a few questions over at Biology.SE from askers who learned about a bit of AI in their CS classes and now want to solve the big open problems in biology without taking the university-level courses in that area. They're a bit frustrating to deal with, and come off exactly like this physicist does: https://xkcd.com/793/


text:
Graduate admissions: applying to the same programs twice
A few years ago I applied to several engineering graduate schools in the US. I received some offers and interview invitations, but I was not able to attend due to work-related and personal reasons. I am interested in whether it would be wise to reapply to some of the schools that I applied to previously. There are several situations to consider:

An application was withdrawn before a decision was reached.
An offer was received and rejected. Some of the universities asked me to explain why I rejected their offer, and I provided brief explanations (the process was standardized - I merely needed to fill in some forms).
An invitation to an interview was declined.
My application was rejected by the admission committee.

I would like to understand whether I could find myself at a disadvantage if I reapply to some of the same programs.
I have seen similar questions here, but the answers did not cover all of my concerns:

Apply to the same University after a rejection
Reapplying to a previously accepted Ph.D. program, will this be considered rude?

Can you clarify a) how long ago you applied to these programs b) what you did instead of attending one of the programs. By "work related" do you mean you have a job in industry and continued working that job instead of starting graduate school?

@Taw a). not too long ago, you can assume 2-3 years ago, b). continued working in the industry. For the last question see b), with the exception that now I am working for a different company (most of my professional experience is related to the area of my interest for the Ph.D. programs).

There's a good likelihood that the people reviewing your application are going to be the same. So for the cases where you a) decline the interview or b) rejected the offer, you definitely need to explain why you are reapplying and why you are "serious" about it this time. This should be part of your statement of purpose. As for the application you withdrew, it's not clear what stage of the review process they were in. To be on the safe side, I would say you should do the same thing for that school, making the assumption they had already reviewed your application and were probably going to extend an offer. So those 3 situations should all have similar sections in the SOP addressing why you declined/withdrew.
For the school you were rejected from, you probably want to say something a little different, explaining that you never started graduate school, and want to start now.
It's definitely not uncommon to apply to the same graduate program twice, and I don't think it puts you at a disadvantage.  If you were accepted once, you will probably be accepted again, provided you explain why you declined the first time.


text:
Mathematics GRE for graduate engineering admissions
I would like to apply for a Ph.D. in engineering in the US. My area of interest (control systems/dynamics) is closely related to (applied) mathematics. Would a reasonable but not outstanding result (e.g. 70%+) in the GRE mathematics subject test give me an edge during the admission process?
To be more specific, I would like to understand how relevant is mathematics subject test for admission to (mathematics-heavy) engineering programs (subject tests are usually not required for admission to graduate engineering programs) and what would be considered a high/low subject test score in this context?

I read this Q&A before posting. I believe that my question is far more specific. The only useful information that I could infer is that the admission committees/lab leders take many factors into account and that "In some fields, the Subject GRE is given significantly more weight than the General GRE, and so a low Subject Test score can be more harmful to a borderline application." My own question is about how relevant is mathematics subject test to some of the mathematics-based engineering disciplines and what would be considered a high/low subject test score in this context?

https://academia.meta.stackexchange.com/questions/1560/what-should-we-do-with-the-can-i-get-into-x-program-with-3-xx-gpa/1563

Check the website of the department you are applying to, and whether they require the Mathematics subject test. If they do not mention anything about it, then it is not required, and will not be taken into account even if you submit it.
As you are applying for an engineering department, I am fairly confident that the Mathematics subject test is neither required nor will be taken into account, even if you write it. It is simply not relevant. (I was in the same boat as you, and have been accepted into the Control Systems area of an Electrical and Computer Engineering department at a well-known public US university. My Mathematics subject test score was neither required nor considered.)
Of course, there is nothing wrong with writing it as such, it just will not be taken into account, whether your score is good or bad.

Thanks. However, how confident are you that your score was not taken into account? Is this something that your advisor explained to you? Did you get a chance to be on the admission committee since you started?

@human The websites of most places that I applied to mentioned that a subject test is not required. When phrased that way, I assume that the admissions committee would not take into account any subject test score I submitted. Further, most engineering applicants would not submit a subject test score, and all applicants are evaluated (fairly) uniformly in terms of requirements.

This makes sense, the last sentence in your comment convinced me.


text:
Potential side effects of submitting several applications within a single university system in the US?
I'd like to understand if there could be disadvantages to submitting applications to several graduate programs within a single university system (e.g. Texas A&M, University of California, etc)? May there be any cross-communication between different campuses/universities within a system during the admission process? Are the admission decisions always made independently? If my application is rejected by one university within a system, is it more likely to be rejected by other universities within the same system? The subject is engineering, but I am not sure how much this matters.

"Always made independently" is too strong. There might be some cross communication, but I doubt that it is common or determinative. You are evaluated for admission by some committee from a specific department at a specific campus.
There would be little reason for a faculty member at one campus to need/want to comment on an application to another, as they wouldn't be dealing with you in any case.
And, for most systems you can't automatically transfer between campuses anyway, so the decision is local. It is even likely that the "standards" differ slightly at the different campuses in a large state system. And some of them are in competition with one another, and not just in football.
So, there are unlikely to be any side effects at all.

To the extent of my experience, in the U.S. it would be a violation of privacy laws if there were any of the sort of inter-campus communication you suggest. Even in the "bad old days" when there was waaaaay too much under-the-table chatter, it was not between campuses of "the same" university. I know of not even one instance of this in Minnesota, in math, since 1982 (and, to be clear, I've been involved in grad math admissions, and/or been the Dir of Grad Studies in math, for all those years...)
If anything, I'd suspect that various campuses of the same university systems might be more independent of each other, since they do compete to some degree.