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How to apply to do a masters thesis in the US or Canada when departments don't post specific positions?
I am enrolled in a Master's program (Computational Science and Engineering) in Germany and currently I am looking for a Master's Thesis abroad, preferably in the US or in Canada.
Unfortunately, I am not sure how to apply for a thesis and to be honest I am walking around in a vicious circle.
What I've done so far:
My first step was to visit the international office at our university. The lady there told me that I should have a look at some university/institute websites, look for specific research areas/topics of my interest and write application letters to the respective persons.
Additionally, I went to one of my professors who gave me a list of people that he has worked with in the past and which probably might have interesting topics available. However, after carefully working through the list and writing down what looked interesting to me he told me that there's a 99% chance that my application will be ignored or denied if I don't explicitly describe what work I want to do. So actually his attempt to help was not of much use to me.
My main concern is: How can I know what I want to do if I don't know what is being offered? Most departments do not explicitly offer positions, in particular I haven't found a single offer for a master thesis at US/Canada department sites.
My area of interests involves fields such as the numerical treatment of partial differential equations (e.g. finite elements, solvers for hyperbolic problems...), numerical linear algebra, etc.
However, if there is no disclosure of potential necessities for a master student I don't see a chance to describe my interests more specific than that. On the other hand, I am afraid that my applications will be interpreted as "bulk letters" if I just write down everything that I would like to work with.
I also thought of reading through some publications/papers of the respective persons, which would be very time consuming. But regarding the fact that I can have at most a slight insight into the respective topics and that I don't even know if there is any need for further work at a master student's level I don't think it is reasonable to spend too much time with that without a gleam of any success.
I would appreciate if someone could help me with some hints on how to come across that. It would be even better if someone who faced the same situation and succeeded could share her/his approach.

You need to tell why you must do your master thesis in US/Canada.

I did part of my own Master's work in the US (coming from Austria).  In my case, I arranged to do a thesis with a professor at my university who then offered this possibility (it may be worth noting that the topic of my thesis was really only decided upon after I came to the US).  I think this kind of “unofficial” path may often be easier than to apply on your own.  That said, my current boss (back in Austria) does take unsolicited applications from abroad seriously also for prospective Master's students, and has invited applicants for interviews.

@scaaahu Besides personal interest there is no actual need for doing it there. I want to broaden my cultural horizon and have the opportunity to see how research works in other countries. But doesn't that hold for almost every exchange activity? I mean, there is also no "need" for doing an exchange semester, still a lot of people do it. But I will think more about it, thank you.

It should also be said that (at least AFAIK) in the US, when people enter graduate school (i.e., the program to get a Master's or PhD), it is not necessarily decided yet whether they will do a Master's, or directly a PhD; and I believe most people go directly for the doctorate (an option which you do not really have in Europe).

Your comment above actually came across to me, such as broaden the horizon, how research is done there,etc.

Make it clearer in your question who do you think will reject your proposal. Your local institution or the one you plan to visit?

My two cents Re "broaden my cultural horizon", as a French student who has spent one year in the US (albeit at the undergrad level), I haven't found it very different culturally. My three months in Japan have been much more instructive in that regard.

in the US, when people enter graduate school (i.e., the program to get a Master's or PhD), it is not necessarily decided yet whether they will do a Master's, or directly a PhD — This varies significantly from one department to another.  It is true that in the US as a whole it is not necessarily decided, but in many US departments, transferring from MS to PhD or vice versa is difficult to impossible.

@CapeCode: My professor told me that in the described case it would be likely that the foreign institution would deny my application.

Not in Mathematics, but I got accepted to a three different Master in Canada, at three different universities: at the first I just talked to people there, several interviews and proposed a four page thesis subject , at the second one I just had a ten minute phone call and a general direction on what subject I was interested in before I was promised that my application would be accepted and at the third one, I had to write a half-page letter of motivation. They all check my grades though. As long as you have the necessary grades, except in medicine, law or engineering, they'll accept you.

Master's have a different meaning in North-America than in Germany. While the Master's thesis typically closes the university studies in Germany, it is considered 'graduate studies' in North America (the Bachelor is considered the 'university degree' and the majority of students join the workforce with it). Masters are either specializations or given to graduate students who wish or have to stop graduate school before completing a PhD. That is the reason why you didn't find any offers for 'Master's theses' on institution websites there.
You approach of asking a local faculty for contacts is probably your best shot, don't give up. Narrow down to a few options and write them an email, explaining your skills, interests and field of study. Ask if they can think of a possible project for you. In your inquiries explain the requirements for a foreign Master thesis host at your university. Stress out the fact that you are merely asking for the right to use the local facilities and a co-supervision of your work. In my case, a professor told me about a project he had in mind, and I expanded it in a short proposal that was accepted by the local faculty and my program.
North-American professors have to worry about funding for their local Master students. You have to make clear that you want to be a visiting student and thus you will most likely be able to waive tuition fees (with some exceptions, like MIT). Along that line, explain that like Masters students in Germany, you will pay for your ramen noodles yourself and don't necessarily need a stipend if it can't be provided (on that note, Canada and the USA require that you prove you have sufficient means to support yourself while on their territory, so start a piggy bank).
With this in mind, there are few reasons for a group leader abroad not to consider your project, since you might contribute to their research while not costing anything. These few reasons include: too much supervision burden already.

Thank you very much for your answer! I didn't know about this particular circumstance regarding the Master. I will try to consider the points you mentioned, especially in the last paragraph. Let me give it a first try before I mark the answer as accepted though.

I had an appointment with my professor last week and he gave me different options. So we discussed about pros and cons and eventually decided that he would write an email to a colleague in the US. The US professor almost immediately replied and seemed very willing to support me with my plans. He told us that he will start gathering information about organizational things that might need to be accomplished. As by now we haven't talked about a specific topic, but as soon as I have more information I will post it here.

@andreee, is everything went well?  let us know what happened after that? I am also in the same situation. Let me know if that works for you.

I recently applied for my final Master's thesis at institutions in the US and got accepted at the MIT. Without having a warm contact such a professor giving you an introduction to some former research collaborators abroad it is definitely a more challenging task. As I didn’t have these strong ties in the research area of my interest, approaching the research faculty was the option I chose. It is important to note that writing your thesis in US/CA is formally being a visiting researcher / visiting scholar. That’s what you should refer to in your application.
In general what you want to do is to draft tailored messages to members of the faculty and show that you are interested in THEIR particular research AND why your could be valuable for him.
In general my application approach was the following:

Find a research area of interest: I identified a broad research area
that I am passionate about and a few sub branches that would provide
room for thesis topics.
Choose some preferred universities: For the
first application round I chose 3-5 different Universities in the US
and UK. I was planning to do the following steps iteratively but got
a positive response in the first round.
Identify research groups: I browsed through the websites of the different chairs and research groups of the preferred universities to find out which group was
working recently on topics in the research area of my interest.
Mainly the section of publications combined with Google Scholar and
current projects they are working on were helpful here.
Find a topic: Based on their research, I came up with rather broad topic.
Inspiration can come from “future work” section in their published
papers.
Identify potential advisors: Within the research group I
looked for specific persons working on the topics. Usually, there is
a staff website or a list of names on published papers. As
professors are usually super busy, my main focus were PhD students
or Post Docs, who usually are also the main authors of the papers.
Hand-craft an individual email: Faculty members at reputable
universities get hundreds of emails every week from students all
around the world wanting to collaborate. So the message really needs
to be individual (not a mass email) and short (quick to read). I
tried to show that I familiarized myself with his specific research.
Aspects of the message were mainly motivation, the research
idea/topic and how I can contribute. As already mentioned your
chances increase significantly if you state that they don’t need to
fund you.

Of course this is not an easy process and you might need a few tries to get a response. But if you convince them about the value you can create being a free research support is a good value proposition. Once I got my acceptance I looked for supportive professors in similar / adjacent research areas at my universities who would possibly supervise it.

Astroturfing is frowned upon here. I've removed the link from your (quite comprehensive) post.


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Should funding be listed under "distinctions and honours" on a PhD application?
In one of my Ph.D. applications, there's a section where I must "List any academic distinctions or honours or paricipation in olympiads etc." I don't know exactly what this means.
Would this include:

A scholarship I earned to study a semester abroad at an (important) university.
The funding I received to attend a summer school abroad? (I didn't present anything, just attended the school).
My current MAs scholarship.

Typically, when you put together a C.V., there is a place for listing the funding that you have obtained, and these would go in that section (unless they are primarily about the honor and secondarily about the money, in which case they would go in the funding section).
For the application, if there is nowhere to list these types of funding, then I would list them in this "distinctions and honors" section.  These are significant pieces of your history, and should be listed somewhere in your application, and if this is the best fit, well, why not there?

I have all this written down in my CV, but I don't even know if there's a section to upload my documents (?). I'll follow your advice. I just wanted to know if I wasn't going to be totally off-topic.

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Should a Statement of Purpose cover Purpose or Capability or Both?
I am applying for OMSA degree from Georgia Tech and the application requires me to submit a 4000-character long Statement of Purpose.
My question is this:
Should a SOP cover my capability and show me as someone who can clear the course, or should a SOP cover why I am interested in the degree and how it will help me in my career?
This is what the website says:

Personal statements

Goals and career plans, and reason for applying to this program. Up
to 4,000 characters allowed for response.
Other graduate programs to which you are applying
Statement of purpose

some1 — Oh god.  Kill this habit now.  Burn it with fire.  Nuke it from orbit.  If this creeps into your statement of purpose, you won't get in anywhere.  The word is spelled "someone".

Your statement of purpose should include exactly what they request: "goals and career plans, and reason for applying to this program".

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What should you do when asked by the chair to raise course grade distribution when the students do not deserve it?
I am an adjunct in a reputable North American university. It is my first semester teaching an upper-year/beginning graduate-level course.
A student raised a complaint against me to the chair of the department because they did not get the grade that they wanted. The complaint is without merit whatsoever as far as the student's coursework is concerned; however, they did mention that the distribution of the course grade was "lower than all the other courses he has taken" (obviously reeking of entitlement here). The chair took issue with me solely on this.
The distribution was around 15% A, 30% B, 40% C and the rest D. The average of the course was 77% which is a C in this particular school. The chair apparently saw this as a big issue.
The compromise which we arrived at is to raise the grades of the student collectively. The reasoning (or excuse) for this grade adjustment is that the students had poor preparations coming into a tough course. But the actual reason (which the chair mentioned) is because poor grade signals difficulty and hurts enrolment and graduation rate to the college and hurts my career because apparently students will avoid taking my course. The distribution of the grades for each professor is somehow available to the students which is a travesty in my opinion.
I am utterly at a loss. The grade adjustment is utter nonsense. I am moving students who cannot even multiply two vectors together into the C or B range. The school is highly reputable. But yet it seems that the chair is encouraging me to abandon basic ethics in favor of business.
Worse yet, the student made up some completely fabricated lies against me to the chair even including that I have discriminated against them based on race/ethnicity (I am of the same race/ethnicity as the student) and the chair is not willing to say a word to the student.
I wonder if anyone else has faced this "pressure to raise grade" situation before and how you have dealt with it or rationalized it.

I added USA and Canada as tags - please remove one of them if one is inappropriate, or add Mexico or something. Obviously this question relies heavily on a given country's academic traditions.

This is your first try at teaching a graduate level course, are you certain the fault for the low grades across the board lies with your students? There's a single common factor for those grades...

RE: "Worser yet, the student made up some completely fabricated lies against me to the chair even including that I have discriminated against them based on race/ethnicity (I am of the same race/ethnicity as the student) and the chair is not willing to say a word to the student." Are you sure that this is a workplace that you want to stay at?

The flip side of the following: https://academia.stackexchange.com/q/116460/75368

Please avoid writing answers in the comments. Comments are for suggesting improvements to the post or seeking clarification about the question. Answers in the comments will be removed without warning.

How much subjectivity is there in your grading? Do you have to make decisions about the value of "points" awarded? Or is it completely objective - completely? Do you "award" points or do the students "earn" points? How much subjectivity, especially, when an answer isn't completely correct? Judgement call? How much feedback do you give for incorrect answers? Just the points or something more?

I'm utterly at loss as to how you didn't know that grade inflation has existed for 40 years.  (Ok, I'm not at a loss anymore, since you're whole life has been grade inflation and easier courses.)

Factors at play: possibly as a young lecturer, you have too high expectations of the students and compare them to yourself. Possibly as a young lecturer, you are not yet experienced in how to teach students appropriate to their level of pre-existing competence (did they know to multiply to vectors before they joined you? Did you teach it on the course?) Perhaps, the university is not quite as good as its reputation, there is a lag. Perhaps the  university just does not care about the product - education - but only about the money. Maybe the students are lazy/entitled/etc. More info needed.

Can you please clarify whether a common syllabus and/or exam and/or text book is used across all sections?

This is hardly the most important part of the question, but when you say "students who cannot even multiply two vectors together" -- what are you even talking about? "Vector multiplication" is not a defined operation (or, more charitably, it's ambiguous). Are you talking about the dot product in R^n, the cross product in R^3, an inner product in an inner product space, some kind of wedge product in exterior algebra? These are all different things.  Perhaps part of the problem with your students is that you are not explaining what you mean.

How many students were there? How reasonably could you expect their performance to not deviate from the norm?

A disclaimer: I am a full professor of math in a large US public university. I was never an adjunct and, except for two years as a tenure-track professor, always had security of employment. Thus, I have never been in your situation. Nevertheless, I do have some observations and suggestions that might be helpful.

From what is written, the problem is not your fault. For people unfamiliar with math, "multiplying two vectors" is something that math majors learn during their first year of undergraduate studies (in Linear Algebra or/and Vector Calculus classes). For upper division and graduate classes, this skill (dot/inner product) of vectors has to be taken for granted and it is not something that one normally teaches at that level. Lack of this skill suggests that students are unprepared for an upper division/graduate class and have to retake some of the lower division classes. (This is a bit akin to the fact that Calculus students have know how to manipulate fractions, otherwise, they simply do not belong to a Calculus class.)

How is it possible that students are so unprepared? (such lack of preparation is real and I observed it while teaching math classes of all levels in my institution). From my personal observations, it is a combination of several factors: (1) Covid-19 pandemic and remote instruction. As the result, many students have some lost basic learning skills (such as taking lecture notes and preparing for exams). Mass cheating during remote exams and lenient grading made the grade inflation (already present in the pre-pandemic times) much worse. (2) Abolishing some of the standardized tests (SATs for undergraduate students and GREs for graduate students). As the result, for instance, in graduate admissions one frequently has to rely upon some unreliable information (for instance, grades).
Consequently, some of the (otherwise quite reasonable) suggestions appearing in other answers on how to deal with a large number of students performing poorly in a math class, simply will not work. What will work is retraining of the failing students by making them retake some lower division undergraduate classes (such as Linear Algebra or Vector Calculus). The alternatives are either not covering all the required syllabus in advanced classes (and teaching some lower division material instead) or "kicking the can down the road" and passing students who lack basic math skills (which is what your Chair is telling you to do).

So, what to do? You are lacking security of employment and simply cannot say "no" to the Chair (which is something that I personally can and would do as a full professor). Thus, you have to agree to pass students who should be failing the class. However, in your situation, I would ask your Chair for an honest conversation and explain that while you agree with his/her suggestion, you see a potential serious problem having unprepared students moving to more-and-more advanced math classes. For future, suggest "prequalification exams" before students take upper division/grad classes and making students failing such exams retake the relevant lower division classes. If this is not feasible, suggest changing (temporarily) syllabi of some of the upper division/grad classes to allow time for covering the more basic material for unprepared students. Another suggestion is to talk to other faculty at your department and ask how do they handle the problem of dealing with unprepared students.

+1 for addressing some of the specifics raised by the OP, while being realistic.

The question feels a little "ranty" to me.  It reads to me like someone who knows that they are an amazing teacher, who did a wonderful job all semester, and had terrible students who failed themselves, and a corrupt chair who isn't backing them up.  I have no idea what the actual situation is, but you are not presenting yourself well.
From the question, this is your first real teaching experience.  Teaching is hard.  My father—who had a PhD in anthropology and a JD; who clerked for Stanley Feldman and wrote legislation for the Democratic party in the Arizona House of Representatives; who spoke several languages, including Diné (the language spoken by the Navajo people); who was, by all measures, a very smart man—took a teaching position after leaving the House, and was nearly run out of town on a rail at the end of his first year.  His evaluations were terrible, the DFW rate was terrible, everything was terrible.
But he took responsibility for much of this failure and, rather than taking the next summer to work on getting papers out, opted instead to attend a several month long teaching workshop hosted by Princeton University.  He approached his teaching with humility and, by the time he moved on from teaching to administration, was one of the most popular instructors at his institution.  But this came from accepting his own faults and failures, and learning how to improve.
I would very much encourage you to approach your teaching with the same humility.
Your department considers your class to have been something of a failure.  It is possible that your students were unprepared, but it is also incredibly likely that you were unprepared, and that you have a large role in the failure of your students.  Stay humble, reflect on your teaching, and figure out how you are going to do better in the future.

You indicated that this was an upper division / introductory graduate level course.  In most classes of this type, the students who are enrolled are subject area majors.  They are self-selected, and are expected to be strong in the field.  It is rare for the average grades in such a class to be lower than a B—in graduate school, a C is generally as bad a an F, as a lot of funding decisions are predicated on a student maintaining a better than 3.0 GPA (in my second year of grad school, half of the cohort below me lost their funding when they all got Cs in one class—this caused something of a scandal in the department).
It sounds very much like your course is an outlier.  This is going to raise concerns.

That being said, let me address the following question:

In a course that I taught this semester, the distribution of grades at the end of the term was significantly lower than the distributions of other instructors.  I am being pressured by my chair to raise my grades.  What should I do?

The pragmatic answer is "Do what your chair asks you to do."  You are junior faculty (and it sounds like you are not even tenure track), so your employment is tenuous, at best.  Make your bosses happy.  If this is not sustainable in the long run, start looking for other work now.
The idealistic answer is "You should do what you think is right."  One of the principles of academia is a notion of "academic freedom".  While this freedom exists primarily to prevent faculty from being sanctioned for the research that they perform (e.g. you can't fire a faculty member for doing research in a controversial framework, such as critical race theory), it is also there to protect faculty from pressure to change their courses.  At the end of the day, you are the instructor of record, and you can assign grades pretty much however you like.  You might not be rehired, and sticking to your guns is unlikely to win you any friends or supporters, but should have the right to leave your grades as you assigned them in the first place.
You have to decide what is important to you:  your idealistic views of the academy, or your pragmatic need to put food on the table and to maintain good relations with the folk you work with.

In my view you have either misunderstood the question or you don’t know that grade inflation really exists. You should ask yourself: what is the goal of teaching? There are no pragmatic or idealistic answers to this question. There is only one answer: the goal of teaching is to produce learning. Any other answer is a scam.

The goal of teaching is not to put food on the table! Teaching actually has a goal that needs to be achieved. If it doesn’t achieve its goal, it turns into scamming the system to put food on the table.

@MahdiMajidi-Zolbanin I strongly disagree.  Your answer is an idealistic answer:  the goal of teaching is to produce learning!  Sure.  I agree.  But now play out the scenario where you act on that ideal, as an early career instructor who has no political power.  You  might lose your job, and maybe your career.  You don't get to teach anymore.  You lose the ability to put your ideals into practice.  Some people are okay with this---maybe you are, too.  If we take the asker at face value, this seems to be the position they are in.

You are basically saying that living has become difficult, to the point that people have to just survive. When people get to the point that survival becomes the issue, then they no longer need to pay attention to the meaning and purpose of things, they can do anything to survive. To the person who is just trying to survive, the "pragmatist" may sound like an "idealist". You are telling me this is how bad things have become. I don't disagree with that. But that has nothing to do with teaching. We should be aware of this.

Part 1 of this answer is too long and much of it is irrelevant to the question. I suggest moving part 2, which actually contains an answer, to the beginning, and moving a shortened part 1 to the end.

@MahdiMajidi-Zolbanin Grade inflation exists but so also do new instructors who don't have a good baseline calibration of where the bar belongs.

@barbecue If the question were simply "I am a new instructor, and I am being pressured to change my grades, what should I do?" I would agree with you.   However, the majority of the original question is a rant about how terrible the students were, and how corrupt the administrators are.  This make the whole thing feel like something of an XY Problem, and I feel that it is necessary to also address some of the underlying assumptions and biases which seem to be present.  There are other answers which do a good job of addressing the purported question about changing grades.

@XanderHenderson I'm not saying you're wrong, just saying the first half is too long. When the first part is too long, people don't get to the second part. That's why I suggested moving it.

@barbecue Yeah, I get that, but the first part is what I think is the important part.  The other parts are not very dissimilar from the other answers.

Since the goal of teaching is to produce learning and not grades, the grades are pretty incidental. They mean very little. By the way, they're grades, not measures. Measures are tied to objective phenomena whereas grades are relative indicators. They have only instrumental value. If their purpose has already been served or undermined they have no value.

It is my first-semester teaching an upper-year/beginning graduate level course.

You do not have political power in this situation.

The average of the course was 77% which is a C in this particular school. The chair apparently saw this as a big issue.

The situation requires someone to intervene.
The answer is to find someone else in your department who has the political power to intervene. Comply with the request, then tell the powerful sympathetic person your story, and see what happens.
The best-case scenario is that your school has some faculty who are unionized. I suspect you are not in that situation, because in that scenario, many faculty have political power and your chair's actions are highly unlikely.
On the other end of the spectrum is a school where no faculty have the political power to address the situation. That school is doomed; let it die and move on.
I do not agree with other answers here which seem to think everything is doom and gloom and every school has lost its standards. I do agree that some schools are in danger. Most schools are somewhere in the middle.
But your question is a simple political question; when you do not have the power to take action yourself, talk with a more powerful person who sympathizes with your position.

Yes, avoid pointless self-immolation...

I think the answer to your question ultimately depends on your personal values regarding integrity versus employment, as well as any dire life circumstances that are unstated in the question (e.g. if your next paycheck is necessary to put food on the table for your family, and the lapse in integrity is not something egregious like violence, then that would obviously push the dial towards employment).
So I'm not going to attempt to tell you what you should do. But I'll tell you what I would do, and you can use that information however you want. I'm also going to assume that you've taught the course in an effective way, most students would reasonably be expected to have learned the skills that they are lacking. (By the way, that assumption of effective teaching has not been addressed in your question, so I would recommend adding some details about how the course was taught.)
Personally, the only time I might choose employment over integrity would be in the context of dire life circumstances. Barring that, I would turn the dial all the way towards integrity. My goal in life (which I'm sure is shared by many) is to do things that impact a ton of people, including and especially myself and loved ones, in an extremely positive way. The more positive the impact, the better. Integrity is kind of like a half-compass in the journey towards that goal. It doesn't tell me what to do, but it does tell me what not to do. If my continued employment depends on corrupt things, then the employment is useless for achieving my goal. So why would I want to continue it, especially given the opportunity cost of missing out on employment opportunities that are actually useful for achieving my goal? It's like staying in a bad relationship. Yeah, breaking up might still feel painful in the short term, but it's ultimately the only way to get to where you want to be in the long term.
In your particular situation, I would probably write a long email

communicating my understanding of the expectations of the course and asking whether my understanding is aligned with the expectations of the department (to make sure that the whole situation isn't just some misunderstanding about that),

telling the department chair that I will assigning grades that are reflective of the students' demonstrated learning relative to the expectations of the course (keeping a level head and using respectful language of course), and

documenting some brief notes about any students who are receiving a low grade (e.g. student X has a grade of F because they have repeatedly failed to demonstrate learning skills ___ and ___ or any of the more advanced skills of which those skills are components, which consists of over half the course, even though I took reasonably sufficient measures ___, ___, and ___ in attempt to help them learn these skills).

If they confirm that they want me to give student X a passing grade despite the information that I provided, then I would just tell them that I'm not going to do so and start thinking about employment elsewhere. If that decision ends up terminating my employment, then fine, I'll just accelerate the upcoming job search. (I would also keep an open mind to different industries, e.g. if you know a lot of math, but you don't love teaching enough to put up with the low pay and political drama that often surrounds it, then there are other options, especially if you can code or are motivated to learn...)

I would rephrase some of this to focus less on students' "ability" to the students lack of "mastery" or "failed to learn how to."  If classes were only based on inherent "ability" then there would be no point in having classes.   Indeed, assuming that your students are capable of learning the material and that your job is to facilitate that is essential.

@Elin I never referred to inherent ability, but that's fair enough, I tweaked the wording to avoid the word "ability" since its connotation appears to trigger that particular kind of reaction.

Teaching is very hard.  It sounds like you did not pay attention to how the students were doing early on.  If they don't know how to multiply vectors, you needed to either teach them to multiply vectors or tell them to drop the class. It is very hard to tell without more details (we don't know what your syllabus says or how you assessed students), but math education is an area where there is a huge amount of research about effective teaching strategies and as someone teaching, you really need to spend time diving into that and into learning in general.  It's a serious job.
It's important as a teacher that you remember that your students are not you, not even you as an undergraduate. You were most likely an outstanding student, but think back to who were the other students in classes with you.  How did your instructors help them all to learn and all get them to Cs?
It's a learning experience and you sound like you haven't reflected on what went wrong in the class. The fact that you say insulting things about your students indicates that you have quite a bit of reflecting and maturation to do.
What should you do? I'd look back over the semester. Are there some things weighted too heavily in retrospect? Where did you lose them?

TLDR: The Problem is You.
That's a strong expression, so I'll explain.

I am an adjunct in a reputable North American university. It is my first-semester teaching an upper-year/beginning graduate level course.

It's a reputable university.  You're a beginner at teaching.  Beginners in anything make mistakes.  It's more likely you made mistakes than that the reputable university doesn't know its business.

A student raised a complaint against me to the chair of the department because they did not get the grade that they wanted. The complaint is without merit whatsoever as far as the student's coursework is concerned, however, they did mention that the distribution of the course grade was "lower than all the other courses he has taken" (obviously reeking entitlements here). The chair took issue with me solely on this.

And you should have taken that criticism as a signal that there was a problem with your approach to either teaching or marking (or both).  You do not seem open to criticism directed at yourself.

The distribution was around 15% A, 30% B, 40% C and rest D. The average of the course was 77% which is a C in this particular school. The chair apparently saw this as a big issue.

You're an employee, so if your bosses say it's a big issue, then you should treat it as a big issue.

The compromise

Let's remember that word.

which we arrived at is to raise the grades of the student collectively.

You don't have to like this.  It's a compromise and your employers want it that way.  Just do it.  Compromising means you should consider the matter closed and move on mentally, which you do not seem willing to do (not from the tone of your post).

The reasoning (or excuse) for this grade adjustment is that the students had poor preparations coming into a tough course. But the actual reason (which the chair mentioned) is because poor grade signals difficulty and hurts enrolment and graduation rate to the college and hurts my career because apparently students will avoid taking my course.

This is known as being practical and living in the real world.  Those are legitimate concerns.  You are in the business of education and you need to accept those realities.

The distribution of the grades for each professor is somehow available to the students which is a travesty in my opinion.

A different issue, but if you're not comfortable with that you can move on to a different job.  It's how they do it where you work and you're unlikely to change that.

I am utterly at a loss. The grade adjustment is utter nonsense. I am moving students who cannot even multiply two vectors together into the C or B range.

Honestly it's hard to find that claim realistic at graduate level.

The school is highly reputable. But yet it seems that the chair is encouraging me to abandon basic ethics in favor of business.

It could simply be that you're not a match for the institute.  Again, consider if you're suited to this role and/or this institute.  Consider if the problem is you or them.

Worse yet, the student made up some completely fabricated lies against me to the chair even including that I have discriminated against them based on race/ethnicity (I am of the same race/ethnicity as the student) and the chair is not willing to say a word to the student.

Unless this had negative consequences for you (with respect to your employer), you need to mentally move on.  In your career it's unlikely you'll avoid people making negative claims about you.  Get used to it and develop a thick skin.  If this did have a negative impact with your relationship to your employer you may need to move on or find a way to address the problem constructively.
Note that, if your tone in this question was anything to go by, it's easy to imagine you speaking down to students or coming across as arrogant or dismissive of their concerns.  If you had e.g. a student who felt they had (elsewhere in the past) been a victim of discrimination, it would (again) be easy to see them interpreting a "teacher with an attitude" as "discrimination".
This may be a sign you need to work on how you communicate and interact with people.

I wonder if anyone else has faced this "pressure to raise grade" situation before and how you have dealt with it or rationalized it.

No rationale required.  Your bosses made a decision and your job is to implement it.  Period.
The real problem here is that you're not willing to accept what you describe as a "compromise" or let go of your personal anger.  Nor do you seem prepared to accept your role in this could be the main issue.
No one else can solve those issues and you won't change everyone else.

Re "Honestly it's hard to find that claim realistic at graduate level" - I have heard similar stories from people I trust (working in North America). Comprehensives/qualifiers exist in North American systems for a reason.

@YemonChoi I've a STEM background and it's hard to see how you could avoid something that basic for an entire undergrad course and still get into graduate school.  I suppose money talks is a thing, but if a student's family has enough money to get away with that level of incompetence, it's hard to see how a first year teacher could have enough clout to do more than accept the compromise offered and get on with it.  We all have to learn to walk away from fights we cannot win, and stand our ground when there's a reasonable chance to win.

@StephenG-HelpUkraine OP may have had different levels of achievements in their studies. I can vouch that I come from a generation where we covered more early on in a year (and it felt completely natural) that is covered today in a full degree. Objectively. The standards have slipped drastically. "Honestly it's hard to find that claim realistic at graduate level." - actually, this is unfortunately a very plausible claim. The question is what to do with it. As it looks, one has to compress the actual scale of achievement in the "departmentally acceptable" grade interval.

"Your bosses made a decision and your job is to implement it. Period." I heartily disagree. Consider the extreme case of a teacher in medical school experiencing pressure to increase the grades. The result is poorly prepared doctors going out into society and costing lives. You have a responsibility to society, and to your students, not just to your boss.

@PeterBloem  In such an extreme case (your words) the responsibilities are a little different.  It doesn't alter the general advice for non-extreme cases and, even in such an extreme case, someone who wants to remain an employee probably doesn't have good odds of being able to avoid implementing the policy without negative consequences.  Typically you'd be looking at resigning and maybe becoming a whistleblower is your extreme case. In the OP's case I'm not sure what is accomplished by such drastic actions over someone who cannot multiply vectors.

@StephenG-HelpUkraine I guess we all have to decide for ourselves what the impact is if our students don't learn what they need, and still go into society as experts. But bear in mind that bridges are built on multiplying vectors.

I'm former military, and used the G.I. Bill to finish my college education. As such, unlike some other students who I kept hearing chant the mantra, "C's get degrees", I was there to actually learn. After all, I was paying for the education, both with my service, and with my own money where the grant did not cover.
In classes, I would strive for perfection on tests. While almost every class had a curve involved, I almost never benefitted. I think 3 classes I actively benefitted from the curve, with two bringing me from a high B to an A, and one bringing me from a high C to a B. The only reason why I didn't have a near perfect GPA was prior college in my youth where I did poorly like Steve did.
Now, at my college, not all of my classes were curved, and not all of them were curved favorably. Specifically, none of the math courses were curved at all, and Physics 2 for engineers was actually curved against the students, only allowing a certain number of As, Bs, etc. in each, no matter how high every person scored. In the end it was a very competitive class. I am left to wonder, what kind of degree program are you teaching for?
Honestly though, that shouldn't make a difference. The grades are a standard. Students shouldn't be there to party, they should be there to learn. Our culture is in part to blame for that. We treat high school as the big end all hurdle, where they can then coast to victory if they get into college. However, the truth is that they have to actually try, or they will as Steve said, be useless in life, and have a useless degree with a wad of debt.
While we do need to change, I'm not sure it's necessarily in the colleges that we need to first change. Change does have to come to there at some point though, but unless our culture of treating college as some sort of party place for teens changes, it isn't going to matter. Other countries have already tackled this problem though, and perhaps we can learn how they did it, if we can humble ourselves enough.
For what you should do, don't be a martyr. Instead of dying on this hill, I recommend trying to engage your students in future classes, and make them more interested in coming to your class somehow. I know that linear algebra can be boring. I got through it by doing autopilot calculations for moving around asteroids for example. Get creative!

Can you explain "G.I.Bill"? Is this something from US?

@user111388 The GI Bill is a an umbrella term for several bits of American legislation which provide benefits to veterans.  Among these benefits are funds for higher education.

Yes, specifically, I refer to the Mongomery G.I. Bill and the Post 9/11 G.I. Bill here in the US. If you were honorably discharged for your service and were recruited under this programs, you received a grant for college tuition and living expenses.

It says that you "voted up Steve Kalgren's answer, as he hits the nail on the head" but unfortunately I don't see anybody by that name  or even an answer that seems to resemble the one you are referencing.

You are correct, the answer is now missing.  Possibly removed when the question was migrated over to educators stack exchange from academia. Maybe he didn't have an account here.  I will remove the reference as it is now irrelevant.

I think you need to realise that the most important thing for grades is fairness. Students who achieve more should get better (or at least equal) grades. And this needs to apply across the board. Grading standards need to be comparable across different instructors, since otherwise the main factor affecting a student's grade would be what instructors they had, not how well they performed. That would be unacceptable.
So it is up to the institution what their standards are. It is not up to you. And they have decided that your standards are too high.
(Grades do not so much need to be comparable between institutions: employers know that they are not, and generally have a good idea of how to allow for this. But you can't tell from someone's transcripts that they just had harsh instructors.)

Don't fight over grades.
Grades depend on so many factors, the curriculum, the ability of the teacher, the ability of the student, the teacher student relationship, whether the student was fit on the day of exam, etc. etc.
Try to to be a good teacher, but try not to care about the grades too much.

I agree that this is where you want to end up, but I don't think it's a good starting point for a beginning teacher. Students care about grades, and with good reason. If they fail, it costs them a lot of time and money. For that reason, they are entitled to care about the grade. I think a well-designed course can help students past caring about the grades, but you need to take that worry away first.

@PeterBloem It did not really get your point. My suggestion was to not pick a fight over grading. Just grade as the dean, or the head of department or the other teachers expect you to grade.

I guess my comment wasn't really specific to the situation in hand. Sorry about that.
More specifically, I think it is important for the OP to care about grades in this specific case. Consider, for instance, what happens if other students complain about the grading being too light, or being ill-prepared for future course, despite getting a good grade. The OP can't say "well, the head of department made me do it", they need to be able to defend their grading scheme. It's their course design, and they need to be able to stand by it.

@PeterBloem The OP can very well say that they graded "following the standards and general grading schemes of the institution". It is not necessary that the OP believes this to be a good way to grade.

The distribution was around 15% A, 30% B, 40% C and rest D.
What is the typical grade distribution for courses at that level at that school?
It does not mean that you should automatically adjust your scores handed out, yet if your class distribution differs much from the college average (which is a refection of students' and other professors' history), then you should be prepared to provide an explanation as it certainly will arise in the future.
Your standards do not need to be in lock/step with the college, yet notable deviations will need explaining.  Considerations to align are worthwhile, yet need not wholesale comprise your integrity.  Perhaps a nudge?

FWIW, I had a noted profession (W. T. P.), that although the class was in analogue circuit design, he tended to grade digitally: many As and Fails.  I still remember his name to this day and think positively of the course. (I did get the A.)

I also had one prof of this kind. He also often used to say before the exam (or even at the start of the course) which grade one would get: "You look stupid, you will fail my course." I got an A and think negativly of the professor.

At a German university, I would never assume that all courses of a particular program are equally hard and have similar grade distributions. For instance, in a computer science program, it's a safe bet that the percentage of students who fail in "Math for Computer Scientists" is much higher than the percentage of students who fail in "Introduction to Programming". I'd assume that this holds for all STEM programs.

@Uwe I would actually be kind of interested to see those statistics.  Both "Math for CS" and "Programming 101" are introductory level classes, but I would guess that the fail rates for the programming class would be higher, since there are going to be a bunch of folk in that class who think they want to major in CS or "computer engineering" or whatever, and they end up dropping out, while those in "Math for CS" are maybe already committed majors, and are both self-selected and incentivized to do better.  But, again, I'd be interested in those stats.

@XanderHenderson Both ""Introduction to Programming" and "Math for Computer Scientists" are mandatory courses for CS freshmen, so the students in both courses are the same. Moreover, "Math for Computer Scientists" may be a proof-oriented course, and that's something that many freshmen find really hard. (The "Math for CS" exam that made it to the news some years ago with a failure rate of 94% was a bit exceptional, though. Yes, it was repeated.)

@XanderHenderson At https://pjahoda6.medium.com/the-first-semester-at-the-best-university-in-germany-ab785d786cfb, a student lists the grade distributions of some courses he took at TU Munich. These are graduate courses, though.

@Uwe Ah.  That is a very different curriculum from what is that in the US.  The typical "math for CS majors" in the US is, in my experience, a 300 level (i.e. a class taken by third year students) math class in discrete mathematics, which typically requires calculus as a prereq.  It sounds like the German system does something else.

I don't wish to reinvent much of what's been said, so I just have one thing to add. There is a basic idea that I think should hold for most classes. If one person makes a mistake, it's their fault. If half or more of the class makes a mistake, it's now my problem.
As the numbers state, most students (55%) made less than a B in the course. Even if this was arrived at in a "fair" way, I would ask myself why this is. Yeah it's true, not everyone can make an A in the class, but education is about learning, fundamentally. If students can't multiply vectors, it is your responsibility to ensure they can multiply any vector you give to them.
Sun Tzu once said something like if your men don't listen to your orders, they either don't want to listen or can't understand you. In the former case,  they're punished. In the latter case, it's your fault because directions should be clear enough for them to understand. In this analogy, it seems like you're the one that needs to make things a lot clearer.
This doesn't mean that everyone should make an A and only 2 people make Ds, but self reflection and critically engaging with what students are telling you will make a world of difference, and I don't care where you're at; MIT, Cal Tech, the university you're at doesn't matter.

Indeed. When others don't do what you directed (e.g. your children don't do X), there are two choices: they are unwilling to do it or they are unable to do it. How one should respond are quite different. Here if the students are unwilling to do the work they should get the grade they deserve. If they are unable to do the work, but were let into the class anyway by the department, should they be punished? The chair has stated their interpretation.

Two things: First, in the current situation, if the chair insists, just request a written and signed order from the chair to change the grade, and, if the chair provides it, do it and consider the incident settled. If he fails to provide it, just tell him that you do not carry out requests that the requester is not serious enough about to put in writing.
Second (for the future), if you do not trust the class (I normally trust graduate students that I've taught before only), make sure that the class requirements and the grading scheme are crystal clear in your syllabus. In that case, you'll be able just to say to the chair: "Well, as far as I'm concerned $65<70$, and that determines the grade, but we can take all student exams and quizzes and go over them together to see if $65$ should be $71$". Note that your syllabus is a legal document and nobody, even the chair, can override what is written there without a really good reason.
I would certainly neither change the grade just based on an informal oral request, nor say that I would not change it under any circumstances. I would just say something like "I believe that, based on the student performance, the grade is correct, but let's double check together; here are all records".
At last, just remember, that not every teacher is for every student and not every student is for every teacher. So, whether one person, or one half of the class fails, it may be just an incompatibility issue. It is true that as a teacher you have more responsibility to try to adjust to your students than your students have to adjust to you, but they still do have that responsibility unless they all have the same abilities, level of preparedness, and learning style (which I have never seen in the class of more than 2 people, though I was blessed once with having such class with 2 people rather than 1).
Just my 2 cents.

This is a long-term solution, so it won't help in the current problem, but I think it addresses a part of the problem.
Develop a detailed system and philosophy for setting a grading scheme
Let's call the raw amount of stuff the students got right the "points". Your grading scheme is the way you translate the amount of points to a grade. This is by no means a simple matter, and it's something that students, and in some cases even higher-ups, will complain about.
If you don't want to spend a lot of time and energy on these arguments, it helps a lot to have a strong philosophy for how you decide the grading scheme. This helps you to be firm when you need to be firm, and lenient when you have been to strict.
To set up and troubleshoot your grading scheme, here are a few things you can look at:

The distribution of grades compared to that of other courses You could look at bar charts, or just the pass rate. This is tricky: if it's off, it can be for many different reasons. Maybe your course is too difficult. Maybe all the other courses are too easy, because of political pressure to deliver good grades. Maybe you're not teaching well (yet). Maybe the students aren't used to your style of teaching. Maybe the preliminaries aren't well taught. Or, maybe the grading scheme is just too strict. In short, it's dangerous to blindly adjust the grades to fit the average distribution, but it could be justified.
Random guessing and other baselines Assume for the sake of simplicity that you only evaluate with 4-answer multiple choice questions. In that case the students will have an expected score of one quarter of the points. You are justified in making this the minimal score, and having it correspond to the lowest grade. Likewise, the maximum number of possible points should correspond to the highest grade (all grades should be attainable is a fair maxim). In practice you probably evaluate in a more varied and complex way, but this basic principle can still be held up. In between these two extremes, a linear scale probably makes most sense, since every point should correspond to an equal amount of work.
Students' score on key items This is a very crucial tool. Look at the learning goals, and decide some key tests that should decide when a student should fail. About half the material should be key items, that everybody in your field should know, and their mastery of the rest determines where they fall on the scale between low pass and high pass. You can either mark on key items explicitly (and fail people who miss too many key items), or just use the key items to check whether the grading scheme you've set is correct. In the latter case, most students who fail should miss many key items, and and most students who pass should answer almost all key items correctly.

I think the last one especially can help you in future discussions like this. In fact, you're applying it already when you say "I am moving students who cannot even multiply two vectors together into the C or B range". This is a key item analysis. The only problem is that you're applying it after the argument has broken out. This will always sound like you're justifying your own grades after the fact.
If you set out these key items before the start of the course, this gives you a good foundation for a constructive discussion on the grades. People (students of faculty) can either argue that your key items are poorly chosen, or that they did hit the key items, so they should get a pass. If neither is true, you can hold fast.
Your job is to test students' abilities. If you let them pass without sufficient mastery, they will go out into society and build a faulty bridge, because their diploma suggested they knew how to multiply vectors. That means you are not just in your rights resisting pressure from the chair, you have a responsibility to do so.
You also have a responsibility to the other students not to let the value of their diploma slip away due to grade inflation. Even to the student with the failing grades: they may be disappointed with the low grade, but they will be a lot more disappointed to find that they spent four years thinking they were studying but not learning anything.
The key to taking that responsibility and resisting pressure, is to anticipate, and to build a strong case before the argument starts. Do your homework, and set up a very sound logic for your grading scheme. For your chair to attack the key items. If he does, and tries to tell you that, as a rule, somebody who can't multiply two vectors together should still pass, then you have a moral obligation to kick up a stink, and your colleagues should support you in this.
Going forward
In your current situation, this may be a good way to repair the working relationship with your chair: suggest that for next edition of the course, you will develop a well-grounded grading scheme, based on a set of key items, and you will put this to the chair for comments ahead of time. You could do the same with a small selection of students: discuss the key items with them, and see if they think it's reasonable. There may be other problems you're not aware of, such as preliminary skills that are missing.
If you have key items and a grading scheme that the chair and students all agree is reasonable, then nobody will have a leg to stand on if a specific student needs a grade boost without doing the work next year.

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Academic with fellowship cannot land a permanent position in her department. Should she stay or move elsewhere?
My PhD supervisor who I consider a friend is hoping to get a foothold on the academic ladder at a presitgious univeristy in the UK. She's held a fellowship for several years now and has a few more to go before it runs out. During this time she managed to build a pretty sizeable research group, is teaching her own course to good feedback and her publication record is on par with, if not better than, other junior academics recently employed by her department. However, despite applying for several junior vacancies at her department where she wants to stay, she hasn't come close to getting one. I can't think of many reasons why other than perhaps her field is not considered sexy enough and the department wants to get rid of it altogether.
I want to bring this up next time I see her. It seems it would make sense for her to jump ship and land a position elsewhere while she still holds the fellowship as I'm sure many other departments elsewhere would be glad to have her. On the other hand that's a bit unfair to her students and it may split the research group. If only a hendful of people agree to go with her, this also means that it would take her a while to get back on her feet after the move. Has anyone been in a similar situation and has any advice? Are there other less obvious reasons why a university would snub someone with a good track record who's bringing funding to the table?

" a bit unfair to her students" Unless she made a definite commitment that she will stay around, this is fair game, and one of the risks you accept when you join a research group (especially with a nontenured supervisor).

@artificial_moonlet She was my supervisor but I'm outside the academia now and I consider her a good friend.

@lighthouse keeper Technically true but they may not be fully aware of the risks. I certainly wasn't thinking about this when I was applying for a PhD.

Given the vagaries of the academic job market, anyone whose contract expires within 'a few' years ought to be applying for anything that comes up. Serious external offers may also focus the current department's attention on whether they really want to lose her.

@Vibex I believe that for any major career decision, it's your responsibility to inform yourself about the potential risks involved. A departing supervisor is a common risk in academia.

This should be her question to ask on here.

If she has a sizable research group and is doing good teaching, it sounds as though she's an asset to her department. Perhaps they are relying on her while using rolling fixed-term contracts to keep costs low (especially if she has a fellowship that pays her salary). Possibly she should be having a conversation with them that amounts (not in this language) to "give me a non-fixed-term position or I'll look for one elsewhere". Ultimately, though, it's probably too specific and individual a question to work well for this site.

Universities in the West are generally quite reluctant to hire anyone into permanent positions who had been graduate students or postdocs or otherwise temporary employees at that same university. This is fundamentally based on the idea that we don't want "inbreeding" of ideas and that it benefits departments and universities to get in people with other perspectives. As a consequence, there may not be merit-based reasons why she isn't hired -- it may all be due to not wanting to hire your own disciples.
The solution is of course to go elsewhere -- as almost all of us had to do at some point. Everyone understands that to be the system, and students need to know it as well. In many cases, one can make practical arrangements where students come along, or where someone who leaves can still advise students at their old university until they are graduated. (I have done both.)

Given that this stated to be the UK and the fellow has been in the position to build a group, the fellowship in question is most-likely a "5-year" fellowship like the Royal Society URF (or similar). These fellowships typically come with the expectation that the fellow is considered for permanent employment.

You are right about this. Universities hate hiring their own grads. I have heard this expressed in a less kind way though. "We taught you, so we know what's wrong with you. It's us!"

This makes sense and I something I haven't considered. I suppose it is also difficult for a department to turn away someone who's got their own funding even if that means leading them on (@mmeent's comment).

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Will a journal blacklist you if you withdraw an article which is under review?
A while back, I (along with my collaborator) submitted an article for publication in an esteemed journal. A few weeks after the submission, I came to know that the journal is an open-access journal and, therefore, requires an APC charge if the article gets accepted.
After the first review report (which was in favor of publication), we mailed the editor to ask if the APC charge could be waived, considering that both of us are students and can not afford such a high value (considering the exchange rate between the countries). We didn't get any reply from the mail, so we paused the inquiry and submitted the revision to the article.
However, we are now worried that withdrawing would cause us a poor reputation and perhaps put us on a blacklist for the journal and its associates.
I would like some suggestions. What can we do? Shall we go along and wait for the second report to come and ask after acceptance whether there is a waiver or not? Or should we withdraw now since we don't have funds anyway? Or should I send a reminder to the editor? What if he says no?

You say "we paused the inquiry". What, exactly, does that do? It, potentially, makes it so the editor thinks you specifically know about and agreed to the APC, because you are no longer inquiring about it. The only legitimate thing to do is to contact the journal/editor and explicitly state something similar (and having all aspects of) "We recently became aware of your APC. Unfortunately, we are unable to pay the APC, because we are students. We request the APC be waved. If you, the journal, are unwilling to wave the APC, then the article we submitted is hereby respectfully withdrawn."

Ah, academia. The only place where content creators not only work tenfold to produce their content, but they have to PAY to get it published.

@Makyen I meant, we didn't sent a reminder after the first mail for which we didn't get any reply.

A reputable journal would treat you fairly in the future, so I assume that isn't an issue.
If you haven't yet signed over copyright or given a license of your paper, then the journal can't refuse your withdrawal. They have no rights to the paper. They wont be happy, having spent some resources, but it happens often enough that it is a cost of business, not a cause for retribution.
In future, you should be more careful in researching venues for publication and understand the issues before you submit.
You should explain the situation to the editor, that you are unable to pay open-access fees and that if they can't be waived that you need to withdraw the article, with apologies.
Some journals will waive some charges in some circumstances. And, once you give a license (or copyright), the journal has those rights.

Ideally, OP should explain the situation to the editor now (rather than waiting for the reviews to come in), so that the editor can recall it from review if appropriate (i.e. if the journal will not waive the fee). It is not fair to let over-burdened reviewers work on something when their work might just get thrown away in the end.

This won't reflect well on you, but it won't get you blacklisted either. You might get blacklisted if you confirm the APC (the journal will probably send you an "APC confirmation" before acceptance affirming what the APC should be), let the paper get accepted (even worse, published), and then tell them you can't pay. You've not reached that point yet, so it's not likely they blacklist.
In the meantime there's nothing to do but wait for a response from the journal. There's no need to withdraw until they say they won't waive the APC, and the onus is on them to not accept the paper while the APC is unresolved.

"transfer" is an option!
An editor may transfer a manuscript under review to a different journal for valid reasons, such as it being more appropriate, an I immagine they could also transfer it to one with a subscription model, without the need to withdraw.
But to answer the question, no you would absolutely not be blacklisted if you simply explain that you can't afford APCs.

It is better to complete the entire submission process and strive for paper acceptance. I do not know if this is the case for you, some journals may implement a reduced fee policy for researchers from developing countries.
Reaching the acceptance stage and later encountering challenges with financing the fees is likely to be viewed more positively than preemptively withdrawing the paper due to fee constraints. This is because you will have completed all scientific inquiries requested by reviewers and the editor. Then, the rest is the administrative part of the editorial process.
It's improbable that journals maintain a blacklist policy. Nevertheless, it's important to recognize that editors are human. I've encountered instances where individuals, who may remember minor details, use their influence to impede the publication of papers presenting competing theories or literature. I don't believe this is the case in your situation.

IMO, this is, at best, disingenuous. They know now that they can't afford the APC. Continuing to proceed, without specifically informing the editor that they are unable/unwilling to pay, when they know that they can't/won't complete the process and pay, as explicitly told to them is part of the process, is, frankly, fraud, as they would be obtaining something of value, the reviews, through implicitly or explicitly giving false information (that they are willing/able to pay the APC, if accepted).

Well disingenuous or not, this is not important. However, I would not say fraud for this. Suppose that at the beginning you do not have any financial issues but in the meantime, you have. Then, why would it be a fraud?
What is fraud is that a Springer or Elsevier journal which makes you pay a bunch of money for desk-rejecting your paper and without reimbursing you. What is fraud is that a journal does not pay either authors or reviewers. There is so much "free" stuff in academia. I do not see anything wrong in proceeding like this.

The fact Springer, Elsevier etc. are profiting without paying authors or reviewers is sad but it's not these companies whose effort would be wasted here — it's the busy reviewers' effort...

Can you elaborate why "will have completed all scientific inquiries requested by reviewers and the editor" will be viewed as positive?  I don't see why the journal would prefer this — but it's good idea to tell the editor they'd be happy to go through the proccess if preferred, as the authors can certainly benefit from seeing the reviews.


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During peer-review, should I comment on the authors' messy code?
I'm reviewing a paper in pure mathematics. A lot of results in the paper depend heavily on computer computations, and the authors have provided in the article a link to the Magma code they used for most of these computations. However, this code is almost impossible to understand due to the messy way it is written. For example, it does not use any indentation, and all variables are given names like 'aaa' or 'X' that do not give any information about their purpose in the program.
On the one hand, the mathematics underlying these computations is explained sufficiently well that it is possible to reproduce the results without using the authors' code (this is what I ended up doing). Also, the paper contains only a link to the code and not the actual code itself, so I'm not sure if the code is really in scope for the review. Moreover, hard-to-read code seems to not be uncommon in academia, and most people don't seem to mind. On the other hand, I think a small amount of work from the authors (who presumably do understand the code) would make this code a lot more usable for others, just by replacing some of the variable names with names that actually convey some meaning.
My question is, is it reasonable for me to tell the authors that their code is needlessly difficult to understand and should be improved?

Relevant question on this site: Should I share my horrible software?

When you say you reproduced their results, do you mean you reimplemented their computation?

Could it be that the software is automatically generated by another tool?

Do you think you could not have reviewed the paper without the code or is it an integral part? I do not mean if you with a good understanding and ability to reimplement it did not need it, but if you expect the average reader to be able to use the paper without the code.

"most people don't seem to mind" - citation needed

How is that code in a pure mathematics paper gets to be particularly relevant?

does the journal have code-style standards that could be cited?

@GitGud This sort of thing has been going on for decades. See, for example, the proof of the four color theorem.

If the authors have provided a link to their code as a reference, then it is appropriate to offer commentary, particularly if the article is based on the code.
However, I would recommend making the critique constructive: offer concrete suggestions for how to improve it rather than just saying it’s “messy” or “sloppy” and needs to be “cleaned up.”

Doesn't this end up in practice just going to be a PR with the improved code? This looks like it's going to offload the work to write good code to reviewers who understand what that means. So people will just keep produce bad code and hope the reviewers will fix it for them. Sound like a wrong incentive system.

@Hakaishin Rejecting the paper is still an option.

@Hakaishin I don't understand this reasoning. aeismail did not propose to clean up the code for the authors and send a pull request, he only suggested to provide actionable feedback rather than just saying the code is messy and needs to be improved. If we go by the reasoning that we can't give any helpful feedback at all, we could get rid of peer review and just do yes/no decisions on papers.

@Hakaishin No. Constructive feedback doesn’t mean “fix the code for you.” The feedback could be as simple as “line up the code blocks better.” The point is that it should be actionable. A vague “fix it!” Helps no one. But yes, if the code is a major emphasis of the paper it’s fair for a reviewer to comment about it.

The code is within the scope of the review, and it is appropriate to review this and offer constructive suggestions in relation to its deficiencies.  Now, bear in mind that the onus is on the author to satisfy reviewers of their argument, and if the argument depends on computer code that is so messy as to be unreadable, it is not incumbent on you to fix this for them.  In this case, constructive advice might be limited to explaining why it is presently too hard to read (i.e., lack of indentation, unclear variable names, etc.), and this could reasonably lead to a recommendation to revise and resubmit.  Try to be clear and comprehensive in describing why the code is presently difficult to read, so subsequent re-submissions can be expected to be up to scratch.
The best thing to do in these cases is to treat the computer code just like the prose in the paper.  Just as with prose, the computer code needs to be clear and intelligible, relative to the standards for coding.  If it is messy and unintelligible then it needs to be revised until it is clear.  Reviewers do not shy away from rejecting papers when the prose is unintelligible, so it is perfectly reasonable to request that computer code be made intelligible.

The code being a mess is probably not a valid reason for a major revision if the manuscript itself merits a more positive recommendation.

If it is too messy to be intelligible, and if the argument in the paper depends on it, then I would say you couldn't accept without changes.  What would be more positive than revise and resubmit in that case?

In this case, the OP verified the results without the code. This is a grey area. Personally I’d mention the issues but if everything else is fixed would leave the decision to the editor.

Yes, you should comment and possibly more.
You've said it yourself:

A lot of results in the paper depend heavily on computer computations.

Well, the program code for computations is therefore part of the work you are reviewing. If the text of the paper was difficult to read, would you not consider that a weakness? Logically, therefore, the same is true for the code (even if it's to a slightly lesser extent).
Also, if the code is unreadable to you - maybe there are errors in it, despite the sound math underneath. And finally, if you can tell what the results should be without the code, then why even have the code?
So, if you feel the messiness does not preclude "parsing" the paper, then comment on it (and perhaps, if relevant, downgrade it from Strong Accept to Weak Accept, although perhaps that's too harsh - depends on the specifics.)
If you need to read the code to verify the results, and you simply cannot, then that's a more serious problem. But before saying something like "Requires revision", consult with the journal editor / the program committee chair / etc.
Note: I'm a Computer Scientist, so my answer might be somewhat biased. On the other hand, I have written pure-theory papers with no code.

Messy code affects reproducibility
You tried to reproduce their results with the linked code, and were unable to do it. While you imply that you were ultimately able to develop your own code and replicate the results, I argue that badly written code affects reproducibility. In computer programming, this can be even more important, as programming languages don't necessarily have very long lives. Who knows if Magma or any other language will be common knowledge in 50 years.
In the long view, reproducibility is the most important part of the scientific endeavour. Proof that doing a results in b, a fact that can be re-proven by anyone who cares to try, is an axiomatic building block upon which further scientific results can stand.
If reproducibility is important, then there is nothing wrong with telling them to clean up their code. Frankly, if their code is as bad as you describe, it sounds like the authors will have trouble understanding their own work going back to it in a few years. In that case, by forcing them to learn a bit about writing nice code, you'd be doing them a favor.

Let me touch briefly on an aspect that hasn't appeared on the existing answers.

My question is, is it reasonable for me to tell the authors that their code is needlessly difficult to understand and should be improved?

Yes, you should comment on the code, but not only that: convince the authors that it is in their self-interest to fix those issues.
Readable code is code that is easy to reuse. Reusable code is code that makes it easy to explore the mathematics presented in the paper. Explorable mathematics are more likely to have readers that find interesting extensions. Interesting extensions get published, and those publications cite the original code - and, moreover, provide some of the most valuable citations around.
Making your code readable and reusable does not guarantee that this will happen, but if you publish unreadable code you're putting up an artificial barrier in front of a reader who might or might not go on to do further research based on your work, and if there are enough such barriers, that reader will just turn elsewhere. Making the code readable is a modest investment of time that results in a large improvement in the extensibility of the work.
This putting-up-of-barriers, of course, isn't unique to code: unclear figures, tangled structure, messy grammar, missing lemmas, and all sorts of other issues can put in similar barriers, and your job as a reviewer includes pointing those out and helping the authors get rid of them. Code is no different - help them improve it!

I'm not in academia or a reviewer of articles/papers at this level (adjunct at tech school), but I do grade a lot of programming homework and the odd technical document sample, and I do software development to actually pay the bills.
If the paper depended on output the code generated, then the code must be readable and understandable - otherwise, the code may not do what the author thinks it is doing and it is impossible for others to confirm w/o their own re-implementation.  If such re-implementation is relatively trivial then it seems that the actual code isn't important, and so I would question why broken code for something that is easy to implement based on spec would be included or referenced in a scholarly paper.
Given that you were able to verify using your own code implementation of his algorithm(s) I don't think that is the case, but it should be taken into consideration.  Any decent IDE or even advanced text editor should be able to auto-format code and do project wide search/replace (refactoring).  Kinda points to sheer laziness....

"Any decent IDE or even advanced text editor should be able to auto-format code" - On the contrary, it means anyone can format the code to their liking. Hence the lack of formatting is not a concern. Sure it's lazy, but it's just an inconvenience for the reader.

@NisargShah, an IDE can't give your variables meaningful names.

I'm a software engineer and I want to answer the question from that perspective. Most code is not readable in itself. You need to have comment to document data structures and specifications for subroutine calls. Academics are not software engineers, and I don't expect them to do a professional job in this respect. Still, it is certainly in order to comment on the quality of the software. Without looking at the actual code, I'm not sure I would comment that it is unreadable, because the article (that is according to testimony, sufficient to reproduce the code) is to be considered part of the program documentation. If the program uses short names, that are the same then in the paper, that is no problem. Missing indentation is not an indication of low quality, but many levels of indentation is.
I would suggest that you express your feeling that you find it hard to read, but that you are no code expert either, and maybe have some software engineer look at it. It is a different skill set you know. That should take the edge off the comment.
To top it off: I've done a good job in cleaning up code, that I didn't understand at the purpose level. You'll be surprised what an expert in a different field is capable of.
Bottom line, the code is not of the essence, the quality of the code is incidental and it shouldn't affect your decision either way.

The authors provide a link in the article, so the code is either consider considered a reference or part of the research.
Whatever the situation, this raises questions:

Is the code archived? Practical ways to archive code include Zenodo or figshare. Code on a homepage is as good as no code at all.
Is there a license to the code? If not, its status is not at all evident.

As a reviewer, it is up to you to decide what to do. Possible actions include:

Do not comment on the code.
Comment on the code with what I would call the minimum: require for the code to be archived and licensed properly.
Depending on the importance of the computer program in the research, require a minimal amount of readability and that the author provides some tests on the program (i.e. that the program provides known analytical answers if some parameter sets allow it, etc).

Regarding archival, you can refer to the editorial information of the Journal of Open Research Software.

The "Journal of Open Research" does not seem exactly trustworthy given that the link to both Codeplex and Google code which have ceased operation quite some time ago. And lastly, one needs to pay for the publication which again sort of raises questions about the trustworthyness, especially given the outdated information and the apparent reliance on authors to host the code elsewhere... (Elsevier also has a "pay to publish" software journal - but they will/offer to host a copy of your code with the paper.)

Their publication fee is quite low in comparison to Elsevier and they originate from a scientific society (the Software Sustainability Institute, so I don't agree with your assessment. (disclaimer: I published with them). You are right with respect to the list of repositories, I will mention it to them.

325 Pounds vs 500 Euros isn't a huge difference. Now they may have a well reputed backer, but this should be properly communicated AND they should possibly keep their resources up to date. Google Code closed at the beginning of 2016 while Codeplex closed in December 2017. (Doing some more googling now, it seems like Ubiquity Press is normally a "recognised publisher" of open access articles - But the journal presentation does not exactly inspire confidence.)

My understanding is the authors do not submit the code for publication, so the code does not seem to be subject to your review. The authors just give a link to their code. This begs a question: if they just had provided a reference to their work published elsewhere, would you have considered telling them that that work should be improved? One may argue that validity of their results hinges on the correctness of their code, but they could choose not to provide access to the code at all, as "the mathematics underlying these computations is explained sufficiently well that it is possible to reproduce the results without using the authors' code". You, as a reviewer, took the trouble to verify the results of their computation, thus going beyond and above your duty, but that means that their results are indeed reproducible.
So I would think that you may mention that their code is lousy (while apparently correct), but that should not affect your recommendation on publishing / not publishing the paper.

If they provided a link to work elsewhere, and following that link lead to a page that was basically unreadable, it wouldn't be a bad idea to point that out to them.

@RDFozz : That does not seem to contradict the conclusion in the last paragraph of my answer.

In my reading of the question, I see nothing that indicates the OP would recommend the paper not be published based on the state of the code (though, perhaps I'm missing something). I was focused on the first couple of sentences of your answer, and trying to point out the difference between a reference to a published work of some sort (which might require actually obtaining a physical copy of the work to look at it, and which is presumably in a format that would not allow simple revision), and a link to a web page (which generally is relatively easy to change/improve).

@RDFozz : There may be some practical difference, but still the code is beyond the scope of review, and the authors may have all kinds of reasons not to touch the code.

I am not aware of any scientific journals that base their decisions on the code. Whether it is messy or not is not a concern relating the journal. That it should be reproducible should be a concern but I am not aware of any journals that make this a criteria, except for some initiative for some communities, e.g. NIPS, CVPR. In my opinion, it is already a bonus that they chose to release it (and in some cases people are not even allowed to release them due to their organization's rules). If you have any claims that the code is not correct or does not do what it is said to do, it might be a different story but you need to justify this. If you only want to make a comment on the sloppiness of the code, I think you should clarify your commentary and emphasize that it doesn't affect the overall review, I agree with a comment made earlier " convince the authors that it is in their self-interest to fix those issues." Just make it known that your commentary about the code is not related to the review of the paper. Also, please don't discourage people who release their code even if it is "messy".