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Is it ethical to study online solutions to for-credit math master's homework problems?
I am taking Complex Analysis for a master's degree in Pure Mathematics. The grade is:

Three written evaluations.

Questions the teacher asks that must be submitted every week.

The teacher's questions serve as supplementary notes. They are difficult for me, as I don't have a good Analysis base. So far I haven't been able to solve this week's question. I have to submit it next Monday. Am I lacking in ethics if I look for a solution on the internet, study that solution and deliver it?
If it were an undergraduate degree I would even understand, but now in the Master's degree I feel like I'm lacking in ethics. How am I going to become a mathematician by acting this way?

Are you allowed to discuss with your fellow students? When we studied, joint discussions were permitted, but it may differ.

As for ethics, I suggest a reframing: if you look up all answers on the internet and find them there, who deserves the Grade A? You or the sources? Assume you are hired on the basis that you got an A in this module, would you get the salary of someone who knows how to google stuff on the internet or that of someone who can solve the problems on their own? Finally, why do you study Pure Mathematics? So that you learn how to do things yourself or so that you learn how to find solutions online? These are the questions that you could ask yourself.

Are you allowed to list & cite sources with your answers?

You need to ask the instructor.

Is it unethical? Yes. Should you do it? No. Is it that big a deal if you search for the solution to one hw assignment on one course off the internet? Realistically, no, not really. But your post makes me worried that you're heading for a whole semester of doing this unless you solve the underlying problem: "[the hw is] difficult for me, as I don't have a good Analysis base." Instead of hand-wringing over copying a single assignment off the internet (not great, but not a big deal in the grand scheme of things), the real question is, how are you going to deal with the above problem?

If you search the internet or books or other places for understanding, then there is no ethical issue. The course actually sounds like it might require that.
To be a mathematician, you need to develop insight. Insight for many comes from working out a lot of problems. It won't come, for most, by reading a lot of solutions.
If you search for answers, rather than understanding, or submit the answers of others, it is an ethical issue in two ways. The first is probably obvious, but the second is that you are cheating yourself out of an education. Instructors don't ask questions to get answers (other than research questions, of course) since they can provide their own answers. They ask questions to initiate a learning process.

As a side note, I found that insight into Complex Analysis was quite different from Real Analysis. I don't know if that is a general perception or not.

Great answer as usual  :)  I suggest adding: the OP should ask their instructor for confirmation that this matches the instructor's understanding; if so, they should also ask the instructor if they should acknowledge their sources. For that matter, they should also check with their instructor about discussing problems with the other students. (I acknowledge that all these ideas appeared in others' comments/answers on this post)

Better to talk with the teacher about how to develop more skills in this area than to pretend they have no issue (which won't help at all when they have the written evaluations).  In general, teachers also need to know what areas people are having trouble with in case they need to modify their approach to teaching or allocate more time for a particular subject.  Teachers cannot find this out if people pretend they have no problems.

I also found insight into Complex Analysis to be very different from Real Analysis. But then I transferred from the masters in mathematics program I was in to law school near the end and I never claimed to actually be any good at mathematics. ;)

There is an easy fix to ensure that you are acting ethically here:
Acknowledge the source.

Eh I don't know about this. There are situations where it'd be unethical to use an outside source at all, e.g. a closed-book exam, and in those cases acknowledging the source you used doesn't make everything okay. I don't think we really know if the OP's situation is one of those. (Though it probably does make sense that being open about what you're doing puts a "cap", loosely speaking, on how unethical it can be.)

So it's ethical to cheat if I write "I acknowledge the person sitting next to me in this exam for questions 1 to 3"?


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Masters research with 3 months left, poor choice of supervisor and topic, is poor thesis with rehashed results be accepted?
From India. I chose a poor supervisor who seduced me with potential progress. I soon realized that the topic was bland, highly theoretical, and only practical in high-end labs. All you can do is simulations in open source software that are just running first order approximations about the model.
I soon grew bored. The supervisor can't even explain stuff. He even "gaslights" us (3 of us are suffering individually) saying how cooperative he is. I almost lost control several times. Three of us agree that he has very poor communication skills. He even threatened one of us saying he will not accept his work. When asked a doubt he finds some paper in 5 minutes by Googling and asks us to read it. He doesn't read the work I sent to him properly.
Until now I have rehashed results from some existing literature and have convinced my evaluation panel. They are satisfied because of my good presentation skill alone and due to the fact the topic is very new. But that was all I did until last semester. I have three more months left. I don't know what to do.
I did try to change the topic and supervisor. I even had a professor who thinks highly of me, someone whose interests I align with ready to take me, but the department head said I can't change topics midway.
I have a dream job at a billion dollar company for a role that aligns with my interest coming up in 4 months. One I only dreamed of when I was a school kid. Now I am afraid that I will lose it.
Will my masters work be accepted in its current state?
I did not bring any novelty, all I did was read about things and present it, and write some code that I obtained from various sources (cited of course). One thing about my topic is even accepted papers in journals are full of grave dangerous assumptions, as a matter of fact, a paper with these rehashed existing results I sent to an international conference exclusively on this topic was accepted!

There are too many variables, mostly unknown, to make a good answer. It depends a lot on the standards for graduation at your place. You might be fine. Or not. A lot might depend on the advisor's judgement and standards.

Strictly spoken, the requirement of originality typically only applies to PhDs. So, an MSc demonstrates your capability of doing research, understanding and applying academic-level methods etc., but not actually finding new things. In other words, if you do the rest correctly, it is not critical that your results are not novel, as long as the methodology is soundly scientific (and that includes appropriate use of literature). I suggest you abandon the "jerk" thinking about people who may or may not have other constraints imposed on their decision, we don't know. Try your best and good luck.

@CaptainEmacs and Buffy Please do not use comments to answer the question!

Sorry to hear of your difficulties, but this is a Q&A site. The only Q here is “will they accept my work,” and only “they” can answer that.

@FedericoPoloni In principle I agree with your point. However, as the question was formulated, my comment was not strictly an answer to the question, so I did not promote it to answer, despite considering this.

@FedericoPoloni Strictly speaking I think you're right, though Buffy's comment mostly explains our too individualized close reason. However I don't think the community wants moderators to be that strict with comment deletion on closed questions that can't be answered.

@CaptainEmacs Ideally in this situation you might link to another Q&A here that focuses on originality expectations in masters theses.

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Organisation of a math thesis
Do you think it's appropriate to introduce notational aspects right after presenting the main topic in the introductory section of your thesis? Personally, I prefer not to include notations in an appendix.
Specifically, what are your thoughts on including very basic information such as notation for real numbers in the introduction? Additionally, do you think it's advisable to introduce the necessary notation for more complex objects in the advanced chapters, at the beginning of each chapter?
Furthermore, would you consider it redundant to explain the organization of each chapter again at the beginning, especially if it was already covered, to some extent, in the introductory part?

Have a look at other dissertations, long papers, short books, and so forth to figure out what you see as effective, courteous to the intended audience, and aesthetically acceptable to you.

Which kind of thesis? Bachelor's? Master's? PhD? And most importantly: What does your advisor say regarding those topics?

Yes, repeating yourself is redundant, but you should do it nonetheless, because it makes the reader's life easier. You are not writing for a robot with perfect memory who reads in a strictly linear fashion, who still remembers on p. 97 the remark you made on p. 13, and who does not understand normal human language and ordinary notational conventions but is a native speaker of first-order logic. You are writing for humans who will typically not read your thesis in a linear fashion and who do not have perfect memory, but who do not need to be explicitly told that e.g. R denotes the reals.

In my experience from reading theses and other long documents, I don't think a list of notation at the start is helpful at all. I'm not going to read the list, and if I don't know what something means I'm more likely to skim upwards looking for a definition than scroll all the way to the top to look at the list. Including things like ℝ on the list makes it even less helpful, because it just makes it harder to find the things that aren't obvious. Instead, just introduce notation at the point where it's used, and make sure the definitions are clearly marked and easy to find.

Advice

Talk to your advisor.  At the end of the day, your goal is to complete a degree.  The primary obstacle[1] to completion is your advisor, followed by your committee.  If your advisor has a strong preference, do what they tell you to do.[2]

Read other theses in your area.  Particularly from your institution.  See what other people in your department have done in the past, and what other people writing on similar topics have done in the past.  Your institutional library should have a collection of past theses by people at the  institution—have a look at what other people from your department (and even advisor) have done, and emulate their structure.

Refer to your institution's guides. Your institution will have a guide for theses.  It will (perhaps) tell you about which fonts you can and cannot use, how wide your margins should be, how sections should be numbered, what pages are necessary, and so on.  It is possible[3] that this guide has some guidance about how to organize your thesis.  If it does,[4] do what it tells you to do.

Read texts in your area.  See what other people in your field are doing.  If appendices of notation are common, go that way.  If introductory sections of notation are common, do that.

Do what you like.  Ultimately, at the end of the day, your thesis is your document.  You get to decide about style.  You have expressed a dislike for appendices, so don't include one!  The reality is that none of the above points are likely to apply.  Most advisors aren't going to care about the structure of your thesis, your institution's style guide is likely silent on the issue, and I would imagine that there is no consensus in your field.

Anecdote
For what it is worth, my own solution was to include an index of new, idiosyncratic, or otherwise non-standard[5] notation just after the table of contents and the list of figures.  I then introduced notation in the basic flow of the text, and had LaTeX build the index of notation automatically.  A lot of notation was introduced in my first two chapters (background and introduction), but I continued to introduce notation throughout the document.  I think that this worked pretty well.
Notes
[1] My use of "obstacle" here is a bit tongue-in-cheek.  Hopefully, your relationship with your advisor is good, and they are an active help, rather than hinderance.
[2] For example, my masters advisor very strongly disliked the Oxford comma.  I very strongly disagreed with him.  There are no Oxford commas in my masters thesis.
[3] Possible, but I think highly unlikely.
[4] And it probably doesn't.
[5] There is absolutely no need to define standard notation in your field.  Don't waste your readers' time with that.

Upvoted. Two small remarks: Regarding 1., depending on the location it could also be an undergraduate degree, i.e. the thesis could be a Bachelor's thesis. In 2. I think the sentence "Your institution will have a style guide for theses" is a bit of an overgeneralization. E.g., none of the math departments where I've worked so far had a style guide for theses.

@jochen  By "institution", I meant college or university. I've never heard of a departmental style guide, though I am sure that one exists, somewhere.

Ok, I see. Though, the universities where I've worked so far didn't have a style guide for theses, either. ;-) (Admittedly, I find it a bit strange if an entire university or college has a style guide for theses. How would, say, the people in literature studies, law, and mathematics, and engineering, agree on the same style guide?)

"See what other people in your field are doing." I would add – see what other people are doing and reflect on whether it works and if so / if not, why.

The placement isn't all that important in theory. It is even acceptable to introduce specialized notations at first use rather than collecting it. If you collect it as an appendix, make an early note that it exists at the end. If you put it early, make sure it doesn't interfere with the flow. But a stand-alone section early on is fine.
Standard notations in the subfield probably don't need to be stated if you use them in the standard way. That might even get in the way.
Your university may have a style guide. Your advisor may have preferences. Don't neglect those.

Yes, emphatically, standard notation should not really be introduced, assuming your audience is mathematicians of some sort. And you should hesitate to introduce toooo much non-standard notation, because it'd be an obstruction. Don't think that math is fundamentally "formal" :)  It is by far best to explain new ideas in terms of old/standard, rather than accidentally pretending to be operating in a vacuum.

@paulgarrett: "you should hesitate to introduce toooo much non-standard notation" I wish one could find a way to force all math book authors in the world to follow this advice. My favourite example is a book where the author introduced a separate notation for the imaginary line (something like $\mathfrak{J}$ if I remember correctly) instead of just writing $i \mathbb{R}$. My short term memory is really too small to hold a couple of unnecessary variables and similar quirks when I read a book (or a paper).

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