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When do most non-tenure track teaching positions appear in the job market
I am a math PhD student in Mathematics in the United States, graduating this Spring. I am only seeking non-research positions because I like teaching more than research.
I have been looking at the Math Jobs positions, and most advertised jobs are non-teaching professorships or postdocs. Is this because lecturer and teaching professor positions are posted later, Math Jobs are not as comprehensive for these positions (I found some jobs that were only on HigherEdJobs), or there are much fewer lecturer and teaching professor positions than research-based jobs?
I have found several sources that say tenure-track positions are usually posted in early Fall, but I can't find anything that estimates when non-tenure-track jobs are posted.

I've posted an answer to the question asked here, but there is also a second implicit  question about finding research questions.  Note that most newly-minted math phds in the US do not get tenure-track positions.  Typically, they work through two or three postdoc positions, first.  By the end of those positions, you should have a much better idea about how to run your own research program (and you should have several papers in the hopper).

Also, for what it is worth, the timeline for hiring postdocs is (typically) similar to that for hiring community college faculty.  Those deadlines are usually in the early spring semester (January through March, ish).  Again, they are looking to hire people who are not going to leave the hiring process mid-stream because they just got a great tenure track position.

There is also sometimes the following: a small department (through some unexpected event) must hire at short notice someone to teach.  For example, someone left in the middle of the term.  So, if you are not going on the tenure track, vacancies could occur at any time.

Don't confuse non-research with non-tenure. Teaching colleges, such as liberal arts colleges, hire a lot of mathematicians and do have a tenure track. They may require some research, but most value teaching and keeping abreast of the field much more highly than research.
So, while, as you say, tenure track jobs are most common in the fall, look at them, but separate out the teaching colleges.
There are some research universities (Duke, for example) that have a dedicated teaching faculty that is, in some cases not tenure track, but long term contracts. But they are likely to advertise on the same schedule as the research jobs, because they are driven by the academic schedule. The people I know with such positions in CS (several) do, in fact, publish, but it is more likely to be in such things as pedagogy applied to the field than strictly theoretical work. Look for positions labeled something like "Professor of the Practice" and similar for these positions.
Note that almost all educational institutions value research and publication even if they don't require it be a major part of the position.
Other non-tenure track jobs are just adjunct positions. Part time and poorly paid. You might be able to contact nearby universities and ask what might be available. But it is a poor choice for a career.
Based on your comments, let me add that the current state of mathematics today makes it very difficult to work alone. No matter where you end up look for collaboration and opportunities to share ideas with others. Together you might come up with something. Don't sell yourself short.

I understand that for many non-tenure track positions are undesirable, but I am happy with a full-time position with a much lower salary than a tenure track position. Even if a job only involves 10% research, I don't think I would ever be able to publish another paper, even if I were still being assisted by a research advisor. I got very lucky in that the topics my advisor chose for me were much easier than other people's research, and I think it would be very hard to find another topic of that level that has not already been answered.

I am interested in full-time lecturer and teaching professor positions (which are rarely tenure-track). Do you know when those positions typically appear? Also, how can I tell which tenure track positions do not expect any future publications and just expect to be abreast with math topics?

The "professor of practice" Buffy describes may also be referred to as "collegiate professor" at some institutions.

@AndrewMurdza You're making assumptions here. Some universities give tenure to teaching positions. In the University of California system, for example, this is called a lecturer with [potential] security of employment. There is a professional activity component, but that's not research in the field. One person I know did conference presentations about techniques in teaching large classes, things like Zoom office hours and chat; another wrote a textbook.

Because the question mentions MathJobs and HigherEdJobs, this answer assumes that the area of interest is the United States (these websites cater primarily to American institutions).  No claims are made about institutions outside of the US.
Generally speaking, academic hiring takes place in roughly two waves each year:

Universities and four-year colleges tend to start the hiring process for tenure-track faculty and non-tenure track faculty (including full-time lecturers) in the fall—they want to know who is going to be teaching classes before they finalize their fall teaching schedule for the next year.  Applications are typically due in late November or December, interviews take place in January (e.g. interviews will often be scheduled during the Joint Mathematics Meetings, which generally take place during the first week of the year), and offers are made in late February or early March.
Note that there are tenure-track positions which do not have a research expectation, or that have a very minimal research expectation.  Many large research universities have teaching professorships (e.g. my own phd institution, University of California Riverside, has several "Assistant / Associate / Full Professor of Teaching" positions in the mathematics department, and such positions are becoming increasingly common), the research expectations at many masters-granting institutions are much less onerous (e.g. in the California State University system, you might be expected to publish three or four papers in your first five years, and papers in ed research often count), and there are a lot of smaller institutions, such as small private liberal arts colleges, which have minimal research requirements.
If you are looking for a position where you are likely to work with undergraduates for the entirety of their undergraduate career, and where you will perhaps get the opportunity to teach upper division classes, these are the kinds of positions you are looking for.

Community colleges in the US typically start the hiring process a little bit later.  My gut feeling is that this is because the community colleges have the sense that they cannot compete with the big guys, so they delay a bit to capture the great potential faculty members who were unable to find work at the larger institutions (you don't want to interview someone who really wants a tenure-track position at an R1, and then have them drop out of the hiring process because they were offered that tenure-track position of their dreams).  The application period for community college positions typically starts in the late fall, but deadlines for applications may be as late as March.  Interviews typically occur in the spring (e.g. during spring break), and offers are usually sent out by April or May.
Note that many community colleges do have a tenure process, but this is much less consistent than tenure protections at bachelors-granting institutions (e.g. in it's infinite wisdom, the Arizona legislature does not believe that community college instructors should have tenure protections).

Note that there are also positions which pop up continouously throughout the year—someone may leave a position unexpectedly, creating an opening; or an institution may just have an idiosyncratic hiring calendar.  Shortly after I was hired (as in, within weeks) a member of the math faculty here resigned suddenly, and another died—the department was scrambling to fill these positions in July.

@Buffy Huh... I could have sworn that I had a sentence about that in there.  I'll add it back.

I think it would be helpful for you to add to your answer information about where community colleges advertise their jobs.

@AlexanderWoo  in my experience, the same places that universities do: math jobs and higher Ed jobs (for math), which are mentioned in the question. But the question is about timing, not where to find those listings. That's a different question.

The teaching-oriented tenure track jobs, such as at liberal arts colleges, follow similar schedules as the research-oriented ones, so they often appear in the fall around now. The job searches probably run a little later, on average, than the research jobs, but not enough to hold back and wait for them.
The lecturer-type jobs are more variable, but tend to show up later, often in the spring. One reason for this is that the number of lecturers needed depends on student enrollment trends which often aren't known until the spring. But as you can see on Mathjobs, some places do their job searches in the fall. On the other extreme, some of them will pop up in the summer because of an unexpected need. Since lecturer-type jobs are not tenure-track and lecturers often look at other teaching positions, it's more common for people to come and go and the job market becomes more like the "real-world" job market and more spread out over time.
There are also "clinical faculty" jobs which are also not tenure-track but at the same time are unlike lecturers in that you are supposed to be interested in an academic sense in the science of teaching. Often they'll be in charge of an overall calculus program or have various other administrative roles. I am not too familiar with their hiring schedules though.

When in the spring do most of the lecturer jobs occur?

Each school is different. I'd just suggest keeping up to date with what's on Mathjobs.


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Is it more important to have an accepted, rather than submitted paper, when applying for math postdocs?
I'm a fourth-year math PhD student and am about to submit my first paper. Since this fall, I probably will be on the job market, I am not sure what strategy should I use for submitting.
My major concern is, how important it is to have one accepted paper on CV than having a submitted one?
If it matters a lot, I’d try some quick, but slightly lower-tier journal. If it does not matter that much, I’d try a higher-tier journal at first. Does this strategy make sense?

Talk to your advisor - because it matters specifically what job market you're going into, and in particular whether you're competitive for research postdocs or are more looking for teaching-oriented positions.

As elaborated in answers, "submitted" means nothing at all in terms of gate-keeping, though that, or putting something on arXiv, does mean that you've thought about something enough to write an essay that's dignified enough so that putting it out in public ("publishing", literally) will not embarrass you. :)

My suggestion is that as an early career mathematician, you really want to ask your advisor and experts in your field what level of journals your paper is qualified for. Submitting a paper to a journal to a too-high-tier journal alone could have negative effects on your career reputation... Of course, going too low is also bad.

@AlexanderWoo from my experience, I am afraid that most advisors, especially those got their PhD before say 2008, may find it hard to tell how competitive their students (unless their students are super strong or super weak but in such cases the students don't need to ask at all) are on the current job markets 2020s or even they could get any position in the U.S. at all. My advisor only told me something like this: try your best and I will support you. If you don't get one then stay and apply again the next year --- or you may go to industry.

Speaking as someone who hires postdocs in applied math: if I'm seriously considering you for the position, I will want to take a look at the paper myself, and I would expect that it will be available on arXiv if it has been submitted.  In that case, whether it is already accepted or not won't make much difference as I can make my own judgment of it.

No kidding! For any sort of hiring, I look at the papers directly, assuming them to be available more-or-less publicly, and do not care about the gate-keeping. :)

I am worried that OP may not fully understand the meaning of "if I'm seriously considering you for the position". Most applicants do not get into the stage of being "seriously considered" at all.

Since you talked about hiring postdocs in applied math... I have never heard any applied math PhD with zero published papers could continue in academia (every one I knew in applied math/stat said they better have published papers even just to get their degrees from their advisors).... Do you have any examples of someone with zero published papers but still got a postdoc position in applied math?

@DanielHatton It's often possible to get arXiv endorsement, even outside of academia, by emailing an endorser with a brief explanation and your best paper.

@NoOne I've hired people as postdocs in applied math in the past who had no published or submitted papers. I did know them pretty well at the time, though.

First, as @AlexanderWoo said, talk to your advisor - these sorts of questions can depend on many factors that are specific to your situation.
Second, if you want a generic ranking of possible statuses for your paper, ordered along the axis of how much they would help your job applications, I would suggest this ordering:

paper accepted to junk/predatory journal
paper does not exist (i.e., not listed in your CV, or listed as "in preparation"), or is claimed to exist but is not publicly available
preprint, available from arXiv
paper accepted to a decent, respectable journal
paper accepted to a good journal
paper accepted to a top journal

As you can see from this list, whether an accepted paper is better than a preprint depends on the category of journal. Having your paper published in a very low quality journal may well be perceived as worse than not having a paper at all.
Note also that I didn't use the word "submitted" anywhere in this ranking. From my point of view as a person who looks at job applications, this label means nothing to me. If your paper isn't yet accepted anywhere, I am mostly indifferent to it being "submitted", but I will give you some credit for it existing in a publicly downloadable form (preferably on arXiv). That allows me to look at it and form my own opinion based on the contents, or, in situations where I can't form a detailed opinion for lack of time or because the subject matter is not within my expertise, I'll at least assume that since you posted the preprint publicly and irrevocably you feel relatively confident about the quality of your own work. That counts for something.
In terms of your strategy, yes, it makes some sense to aim a bit lower with the journals when you are under time pressure to get papers accepted compared to when you are, say, a tenured professor who isn't on the job market. But other considerations could also matter (e.g., some journals are known for getting papers reviewed relatively quickly, others relatively slowly, and yet others for getting papers rejected quickly by making heavy use of desk rejections, which can also be useful). Also, different people have different levels of risk aversion: some might prefer a high-risk, high-reward strategy (e.g., aiming for the very top journal your paper can conceivably get accepted at), others prefer to be more conservative. So we can't tell you what's the best strategy to follow. And again, it's a good idea to ask your advisor for more specific advice. Good luck!

Can you explain why "paper accepted to junk/predatory journal" is worse than "paper does not exist"? One could simply post that paper on arxiv without submittion then according to your criteria it becomes better than "paper does not exist" haha

@NoOne My reading of the context (in particular, the commentary about arXiv) and interpretation is that it's a bad look as it implies that the only place that will touch your work are junk / predatory venues; that is, "the author knows it's so bad they're not even trying the legitimate publication process". If this is the case an extra sentence in the answer to make that explicit would clear up confusion on anyone else's part

I think Dan would agree that there is an implicit 1.5: "No mention of the paper anywhere". For many folks, a paper in a junk/predatory journal is worse than no paper at all, since it marks you as a "not serious" person.

@JohnMadden I know the meaning of predatory journals. But what is the meaning of "junk" journal? Is it just a discriminative judgement of journals with low impact factors? I have colleagues calling any journal with IF below 0.5 a junk journal. Do you agree with him?

@NoOne I'm sure there is next to no agreement on what constitutes a "junk" journal, and impact factors vary so greatly by field that attempting a universal demarcation therewith is not productive. If I'm going to attempt some mind-reading, I think Dan is referring here to "paper-mill"/"pay to publish" style journals, which is to say whether a journal is "junk" or not depends not on the impact of its articles, but on whether it has a "real" peer review process in place.

Maybe it goes without saying, but another category could be "retracted paper," as well as "available on arxiv for much longer than the average time it takes to publish," both which would probably be at or below the level of "predatory journal" on this list. I mainly bring that up since I feel like one of the relevant issues in this question is that submitting a bad or incomplete paper just so you can say "submitted" on your CV is worse than not submitting.

@JohnMadden “no mention of the paper anywhere” is what I meant with “paper does not exist”

@NoOne if you have a paper and “simply” post it on arXiv, that sends a certain message about your state of mind, your knowledge and opinion of the contents of the paper, and your future goals for it. If you publish it in a junk journal, that sends a very different message. One is somewhat good, the other pretty clearly bad.

Submitting a paper means little, actually. Lots of things that are submitted are rejected. It implies nothing about an independent review of the paper. Acceptance, on the other hand implies, at least for a reputable journal, that others in the field have reviewed and judged that the paper is worth publishing and presenting to the world.
But, your advisor can say something about where it is better to submit. If they feel that the paper is very strong, and can say so in any letter of recommendation, then it might be fine to try for the higher rated journal.
The "tier" of the journal probably means less than you think it does, as long as both are reputable and known for publishing quality papers. You all need to think a bit about time to publication. After submission it can take quite a long time to reach acceptance or rejection. And, again, the "tier" of the journal may not be obviously correlated with this time. Talk to the advisor.

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Appropriate timescales for an undergrad research project
I'm a sophomore in engineering in India, and I have an opportunity to take up an undergraduate research project to investigate the efficacy of deep learning for medical imaging. This is at a university with a top medical program. However, the data has just started being annotated and due to this, the project could take as long as 2 years (according to the faculty since CT data is pretty voluminous). There would be some periods of very less work, while some periods would need huge amounts of work in a week.
The main point of contact/research mentor would be a grad student, who in turn works under some professor. They are very experienced in the medical imaging field, but have just started venturing into AI. I have the chance to publish as a non-first author if the results are worthy of publishing.
I plan to apply for grad school right after undergrad. The possibility of publishing seems to be a great offer, but if it were to take 2 years from now to finish the project, I may not have a publication to include in my applications. I will probably have an LoR (preferably from the prof) and gained some research experience though.
My main question is: Are undergraduate research projects on such long timescales (like 2 years) likely to stagnate and/or get abandoned?
I'm not experienced with CT scans but 2 years seems a bit overboard to me for training and evaluating models, since most of the delay is coming from annotating the data.
PS: I do have the chance to get similar opportunities with smaller timescales later, but the potential impact of this project is high, and hence my personal liking for it

This seems more like a local question. The answer could vary a lot depending on the local traditions. For some, a single semester would be all that is available. For others, a longer period. Ask a professor involved in the project(s).

@AzorAhai-him- reduced to the main question on timescales

@Buffy I'm from India. Most of the professors who do such projects finish them within a year. If they are sanctioned for 2 or more years, they often publish intermediate results each year. It is possible that this project too has some intermediate results, but the general pace looks dreary to me at the outset

Thanks, I retracted my close vote. Are you being paid?

Undergraduates are not expected to finish projects, just work on them.  Your supervisor should provide long-term management.

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Consulting during PhD?
Say I got a Master's degree in Machine Learning and had 3 years experience as a machine learning engineer and 2 years experience as a software engineer. Would it be possible to do freelance/consulting work during the evening after I've already put my most productive hours into research earlier in the morning and afternoon?
How likely is it that my time would be better spent wrangling data/reading papers or that my program wouldn't allow external work such as this?
How about doing this work over the summer while also doing an internship? I'd like to make a little more than the stipend a year to put in the bank, and I want to work in industry research after my phd.

It's up to you: I actually did consulting during my PhD, but I also used to sleep just 3-4 hours per night.

Is the question really as you are asking if this is possible? Well, more improbable things are possible, so clearly yes. The specifics of your situation, and the type of consulting are both important. Are you already in a PhD and have a concrete oportunity? (asking the question conditional on PhD and opportunity) Are you already in a PhD and consider looking for opportunities but do not have something concrete? (asking the question conditional on PhD only) Are you considering a PhD and want to see what your prospects for a stipend on the side would be? Do you have the opportunity but consider the PhD? Why do you want to get the PhD / what do you think you ll do after (your profile is not suggesting you want to continue in academia after the PhD, am I wrong?)
In any case, keep in mind that the most important factors are:
-If you are already in a PhD and have one: your advisor. What does he think? Can he help you find something relevant to your thesis/ help you get exerience / connections? Talk to him and make sure he is ok. He might also be able to provide good alternatives
-A think you learn with experience: A PhD should not be a lengthy affair. Every year counts and you want to finish as soon as possible. Especially in a field like yours (ours ;)) you can have a lot of opportunities to get paid for doing the PhD... Do you really need the extra stipend if it will delay you? Unless you get something other than money back, that you can use for your thesis or after the phd.
-What you want to do after the PhD. The PhD is a lousy end goal, it can only serve as the means to an end. What is your optimization function? What do you want to get out of your PhD? How is the extra activity going to help you?
-If you have not yet started the PhD yet, and you really care about that stipend, it might be worth noting that there are many situations where the converse might be more productive and easier, especially given your profile: getting a PhD while working in industry, You would have to find a company (more difficult part, they pay...) and an advisor (easier part: they do not have to support financially, and they get money and a PhD student) willing to support you while doing the phd... You (and secondary the advisor) will want to make sure that the company commits in paper for at least a number of years though(i would suggest minimum 3): otherwise its possible a year in the future the company's scope will change /money will run out

Thank you for your response; it gave me a lot to think about. I'm not enrolled in a PhD yet. As for what I want out of my PhD: I want the experience of conducting legitimate research on a specific topic for a few years. I  then want to get a job applying all the transferable skills I've gained. If I can work as a researcher, great! If not, as long as I can do research on the side I'm content.

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How does grad school view a consistent B average in a course sequence?
I'm a math major, and in an undergraduate real analysis sequence I got a B- and B over the two semesters. A year later I took the first year graduate real analysis sequence and got a B+ and a B over the two semesters. I know there's a lot of outside factors to consider, but in general how do these grades affect my ability to get a Master's in Statistics and a PhD in Computer Science?
My overall GPA is an A-. How can I remedy this track record if it looks negative? Does it involve taking even higher level courses in Real Analysis  (i.e., comparable to 6000 level) and getting A's?

It certainly helps greatly, but I doubt you could easily get As if you're getting B's in all previous real analysis courses...

Yeah, it'd have to be the only courses I took that year, and I might need to find a tutor/schedule extra office hours. Getting a would be B to a B+ or a B+ to an A- should be doable

I appreciate your criticism. Do you have more to add, say the resulting consequences or how to improve my situation if that option is not productive?

Instead of bothering with taking more courses, maybe you could work out an arrangement with a statistics or CS faculty and work on a research project in your areas of interest.

Although this might be a showstopper for some specific schools), this need not be a show stopper in general. Especially if you have at least one very strong letter of recommendation.  My answer is based on the assumption that you would apply for CS or statistics, which are just tangential enough to be able to pass muster from most people's point of view.
It could be more harmful if the B sequence were exactly aligned with what you're going to apply for.  But it isn't.
(There are of course other ways to increase your chances, e.g.: don't apply late; if there's a subject GRE required, a strong score here could help; get some feedback on your essay draft.)

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What is an example graduate admission grading system? How much is GPA weighted?
It is said in a variety of sources that the graduate admission committee uses a grading system. For example, if the total score is 10, the adcomm will assign 6 points to research and recommendation, 2 points to GRE, and 2 points to GPA. It is also heard that GPA is only used for the initial screen because different school grades differently.
So my question is, where I can find an example grading scheme? i.e. How much is GPA, GRE, research, and recommendation weighted? I need to know what is the best thing to work on for the grad admission.

"It is said in a variety of sources that the graduate admission committee uses a grading system" - What are these sources? I have never heard of such a formulaic approach for graduate admissions (and most certainly not for PhD admissions - I suppose such a thing might exist for some masters program someplace).

Don't bother looking for a magic formula. Everything that is considered by the committee matters, and it's unlikely that you'll ever find someone who will tell you explicitly a specific cutoff for GPA or GRE that will get you in, because you could have a high GPA but lousy letters of recommendation or vice versa, and neither case will likely result in admission.
Moreover, many schools will not operate on such a system, particularly in smaller departments that do not have huge numbers of applicants. I know that I've never used such a system at any of the institutions I've worked at.
So the best advice, trite as it is, is to try to work on everything. It's more important to find someone who's willing to champion your application in the admissions committee than try to "check boxes." You never know what it is that will get you in—perhaps someone's letter of recommendation resonates because he or she was a classmate or personal friend of someone on the committee, or perhaps you expressed an interest in your personal statement that attracted the attention of someone looking for a new graduate student. (And, of course, a high GPA never really hurts.)


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Is it legal to accept a later offer (e.g., came after 15 April), despite having previously accepted an offer from a university before 15 April?
I have received a really good graduate admission offer from a reputable university in the US but without any funding offer. As I have to accept their admission offer by 15 April and since I haven't yet received offers from some of the universities that I've applied to, so I wanted to know whether it is okay and legal to accept a later offer (e.g., which came after 15 April) with funding, despite having previously accepted an offer (without funding) from a university before 15 April. Thanks.

By the way, you should immediately (and I do mean immediately) write to all the programs you haven't heard from and tell them that you got an offer from [Great] University X but that it is without funding.  If a top program will take you without funding, then the chance that some other program will take you with funding knowing that seems not too bad.  You will feel terrible if a funded offer comes after you accepted an unfunded offer.

Legal, yes!  Ethical, no!  It's like the boy who cried wolf...  If you say you're committing to doing X, and then change your mind to do Y, you are likely to give the impression that your flaky, immature, or otherwise a "flight risk" and this reputation will likely stick with you in the future.

tl;dr: You're asking if it's legal to lie.

@Paul It is definitely ethical (though not necessarily advised) to change your commitment, it just takes some additional hoop-jumping. You can read the April 15th resolution for the specifics, but it lays down explicit guidelines for this situation.

@Roger: First, the April 15th resolution only applies to funded offers.  Second, the OP is asking about reneging after April 15 (it doesn't make sense to accept an offer before the deadline in the OP's situation; clearly he should wait until the deadline to see if he gets a funded offer).  I don't know why you say it is "definitely ethical" to renege on an offer after April 15th.  (If the offer were funded, the resolution you link to says that the student cannot do that without the express written permission of the accepted institution.)

@PeteL.Clark If it's okay to renege on funded offers (which the resolution explicitly says is true, given that you get the proper permissions) then in my mind it has to be true that it is okay to renege on unfunded offers as well. And I agree that one should wait until the latest possible moment to accept any current offers, but it isn't uncommon for offers to come after the April 15th deadline. To pursue those offers, you'd either have to not have gotten in anywhere, reject all your other offers in order to wait on an uncertain offer (which is insane), or renege on a previous accepted offer.

@PeteL.Clark On a side note, I'm assuming that when the OP says "before April 15th" he really means "right before the latest possible deadline to accept an offer that expires on April 15th."

@PeteL.Clark The written permission of the initial institution is exactly what I'm referring to when I say "additional hoop-jumping." The fact that the resolution states that means that reneging on an offer after April 15th for another is ethical, you just have to get the permission (which I assume most schools will provide).

@Roger: I wonder whether we're just disagreeing about the meaning of words.  If I promise you that I'll do something, then I come back and ask for permission not to do it, then if you give the permission I'm not reneging.  (We make promises to each other, not to the gods.  Any agreement can be mutually dissolved at any time.)  Calling the step of getting permission "hoop-jumping" seems strange, since there is a chance that the permission will not be given.  But sure: if they let you out of it, fine.  I said that in my answer.

Finally, I don't at all think it is "insane" to reject an unfunded offer for the possibility of a funded offer, however remote.  Depending upon the field and the OP's situation, that's either a reasonable choice or the only reasonable choice.  For instance, in the field of mathematics I wouldn't advise anyone to take an unfunded offer the first time around.  In my opinion, it would be better to wait another year and apply again.

@PeteL.Clark I think that's accurate. The "insane" comment wasn't so much directed at the OP's situation, but at the very similar situation where you are holding a funded offer but are still on a waitlist for a significantly better one.

They are not paying you, it means that their investment level in you is low or minimal, decline the admission in a polite way and move on...

You ask whether it is "legal", but it is probably not a legal matter.  You can drop out of an academic program at any time with no legal consequences beyond paying the relevant fees.  Unless your acceptance means signing a contract which specifically requires something on your end, I don't see what you could possibly be legally held to.  I am not an attorney but rather a professor, and I can say that in practice no American graduate program would pursue legal recourse against a student in this way barring some truly bizarre, unique set of circumstances.
The relevant questions are rather whether it is ethical, whether it is polite, and whether it is in your own best interests.  The first question seems quite clear.  Are you really asking whether it is "okay" to rescind your word in a professional context?  I would hope that you know the answer.
There may be some situations in which it is so much in your best interests to renege on an offer that it could be worth asking about the possibility of doing so.  Most graduate programs in particular are not insistent on keeping students who have decided that they don't want to stay there.  But if you're asking about accepting now with the explicit plan of reneging later: I don't really know what to say without it sounding like a lecture on adult responsibility.  This is not a specifically academic issue. I can only hope that your life up until now has given you some useful experience.

I think the ethical question is more complex than you make it out to be.  Unless this is a very highly regarded prospective student, the power relationship is almost entirely one way here.

Every year, many offers end up being made after April 15th for all kinds of programs. I don't think it's irresponsible at all to accept an offer before the 15th knowing that you will pursue another offer if you get off a waitlist somewhere else after the 15th. Otherwise, pursuing those offers would be incredibly high-risk for prospective students, who are already in a position of very little power. It just takes some additional hoops, which are explicitly laid out in the April 15th Resolution.

@Roger: This replicates a comment above, but: the April 15th Resolution applies to funded offers only.  In that case, if by "hoops" you mean "asking for permission from the department whose offer you accepted" then that's what I said in my answer.  How much "power" the student has doesn't seem to be the issue: if you agree that you're going to do something, it's unethical to back out of that agreement because something better came along later without getting permission to back out.  It's not "evil" to break your word; it may be what you feel you have to do.  But it isn't ethical.

@PeteL.Clark My point was more about the "lecture on adult responsibility." As an applicant, I think it's perfectly acceptable, on April 15th, to think "I am accepting my current offer at X, but if I receive an offer to Y in the coming days then I will attempt to renege on my accepted offer in favor of Y's, following the conditions laid out in the resolution." I think that's a perfectly moral and ethical position to hold.

@Roger: As  we established above, I wouldn't use the word "renege" as you did, but otherwise: yes, I agree.

I think the power dynamics are more relevant than you’re allowing.  Students weren’t involved in negotiating the terms of these agreements (like they would be in most countries where workers have rights and fair representation.  I’m not convinced that under these circumstances the students word can be considered freely given and uncoerced.

As several other people have said, as long as your agreement is only verbal, this isn't really a legal question, but an ethical one.  When considering whether one can accept an offer like this and then back out with a relatively clear conscience, one mainly needs to consider what you're being counted on to do in the next year, and how disruptive it will be if you don't show up and do that.
It sounds from your description like what you will be expected to do is put your butt in a seat, and sign tuition checks.  If that's the case, I think you can pull out without too much guilt if you get a funded offer.  Will it really be that disruptive to the program if you're not there?  I don't think so.  Obviously you should expect to (deservedly) lose any deposit you pay, but I don't think it's likely you'll have any real negative consequences beyond that.  Certainly, if I were the graduate director at this "reputed university in USA," I would understand your decision.
EDIT: Incidentally, while I understand it is important that grad schools know who is coming to their program next year in a timely manner, I don't have much patience for moralizing at the OP, since the current situation that they might end up in is a creation of the schools, not the students, and one that the schools positioned to fix, and the applicants are not.  We have known for fifty years now how to solve this problem, and haven't done so, even with an extremely successful and smoothly running implementation for medical residents that in many cases the same institutions participate in.   I understand that it's an enormous collective action problem, but until it's fixed, people occasionally flaking out on us to take other grad school offers is a very small price we will deservingly pay.

I agree that if you get in a situation of having committed to an unfunded offer and received a funded offer, it is worth asking if you can back out of the unfunded offer.  As you say, in most cases I think people will be very understanding about this.  What I like less is the idea of accepting the unfunded offer while planning in advance to renege if the better offer comes along.  Really think twice about accepting the unfunded offer anyway!

@PeteL.Clark Fair enough.  I think if you're making unfunded offers in a field where a student has a reasonable expectation of getting funded offer, then you must know that this comes with the territory.  Probably part of this is the context of mathematics, where at least I regard an unfunded offer as something approaching an insult (this certainly does not apply to all fields).

@PeteL.Clark However, I don't understand what distinction you're making about premeditation; does it matter whether the OP had a cunning plan all along or is so excited to receive a funded offer s/he changes his/her mind spontaneously?  Once we agree that there exists some offer for which s/he would renege on the already accepted offer, as the old adage goes, the rest is haggling over the price (http://quoteinvestigator.com/2012/03/07/haggling/).

To me it does matter a bit.  People change their mind over time, and early career graduate students are famously mercurial.  You can commit to attending a school and then drop out one month into the program, and that's probably okay.  What I don't like is accepting an offer knowing at the time that you will definitively turn it down in case of getting a better offer later.  I think that means that there was something dishonest about your acceptance, and it undermines the courtesy that was extended to the student by means of the universal deadline.

@PeteL.Clark So what if the OP tells themself that they is going to stick with the decision, but something deep in their heart tells them that if they get a chance at a funded offer, they'll probably decide to take it in the moment?  What if conversely they just need to tell themselves that they can switch to a funded offer it appears later to move forward and take the offer on the table now?  Mens rea is complicated enough in a criminal context.

I don't know, man, it gets tough.  Like you I am finding the "unfunded offer" hard enough to wrap my head around.  Only with restraint am I not saying "Hey, never take an unfunded offer.  Why do you think you'll get funded later when others around you are getting funded now and you're not?"

It is possible, legal, and even ethical to do so, though there are some additional stipulations that you will need to follow (see below). Of course, you still run the risk of burning bridges with the people from the school that you end up declining, so think and plan carefully with that in mind.
Most (North American) universities are signatories of the "Resolution Regarding Graduate Scholars, Fellows, Trainees and Assistants" by the Council of Graduate Schools (commonly known as the April 15th resolution), and as far as I know even those that aren't signatories generally follow it's guidelines. You can read the agreement here.
The rules regarding funded offers, including offers after April 15th, are explicitly stated.

Students are under no obligation to respond to offers of financial support prior to April 15; earlier deadlines for acceptance of such offers violate the intent of this Resolution. In those instances in which a student accepts an offer before April 15, and subsequently desires to withdraw that acceptance, the student may submit in writing a resignation of the appointment at any time
  through April 15. However, an acceptance given or left in force after April 15 commits the student not to accept another offer without first obtaining a written release from the institution to which a commitment has been made. Similarly, an offer by an institution after April 15 is conditional on presentation by the student of the written release from any previously accepted offer.

As far as I know, most schools will (not necessarily happily) give this permission if asked. No one wants to force you to go to their program if you don't want to. But it does make for some awkward at best interactions, and remember that academia is a very small and connection-driven place.

It seems worth noting that the whole problem is that the first school didn't offer him funding, which would make this point moot.

@BenWebster I think that, ethically, if it's okay for an initial funded offer (which it is given the conditions stated above), then it has to be okay for an unfunded offer as well. Of course, there's the other side of that situation, which is whether it's okay to drop the unfunded offer without asking permission, which you are correct the resolution doesn't address.

Many departments (at least in mathematics, but I believe also in other fields) have agreed not to "poach" each other's students after the April 15 deadline.  If such a department, say at university X, wanted to make you an offer (funded or not) after April 15 and if you've already accepted an offer from university Y, then the admissions chair at X should first ask the admissions chair at Y for permission to make that offer.  I would expect that Y would grant permission if Y's offer was unfunded and X's planned offer was funded, but there's no guarantee about that.  (When I was graduate admissions chair, in the late 90's, I was once in the position of X, and Y gave permission but was not too happy about it.  In that instance, both offers were funded.)

I've been in such a situation and I know it can be a huge dilemma. I also asked a similar question here.
I believe that these situations are very common within academia. And usually, I think that there are two schools of thought. People that live within Academia and insist on the ethical part of the situation, and people that focus on your very personal interest and your particular situation.
This answer is both an answer to you and myself. I agree that there is something unethical in such a decision. I don't really know what future implications will be - people in academia really like pointing out that academia is a small world where everyone knows everyone and you will damage your reputation. I don't really believe that this is true. What kind of people are these, who will remember you for a lifetime and isolate you, for changing your mind about a decision a few years ago?
It is about your future. These problems begin when you have too many choices, too much information. Today, I know X, so I'm planning to do Y. Tomorrow, I know Z, so I'm planning to do something else. This is how it goes, and timing is never perfect. At the end of the day, I think that both the schools and the Professors will find their way, either with you or without you.
And I don't really like people scrutinizing into your intensions, trying to adjudge if you were saying the truth or not at the very moment you were accepting the first offer. Honestly, ethical lectures make me angry. In my book, it doesn't matter if you quit because you are sick, you broke your leg, your cat died or you received a better offer. The result is the same. Let Saint Peter judge the rest.
Good luck

EDIT: I've been awake for far too long and missed the part about it being GRADUATE, not undergraduate. (However, if you're not the original poster and you're applying for undergrad, go ahead and read the rest!)
If you think you might have to reneg on a graduate acceptance, considering the increased effort applied by the college and the possible professional implications later on, it would probably be best to talk to them about it first. At least let them know that it's problematic for you to accept their offer so soon without funding, when others may offer you funding that could make a huge difference. If they cannot at least acknowledge that this is a reasonable concern, they may not be the kind of people whose cooperation you want to chain your future to.

I agree with much of what Pete L. Clark says, but not all.
The ethical implications of accepting an offer and then reneging work like this: In college, you're a number. If your number goes off the list, someone else's number gets onto the list. By accepting a more preferable school and reneging on the less preferable school, you do a service both to yourself AND to the student who discovers that they've been given your place.
The admissions staff is not going to take it personally and get offended. They will get LOTS of letters from students who accepted, but have to call it off. Maybe they decided to join the military. Maybe they had a family issue and they have to take a second job. Maybe they are expecting a child. Maybe they have an illness. Maybe some other thing out of countless possibilities happened. Some students won't even write to inform them of this; they'll just not register for classes. Believe me - colleges are set up for this because it happens ALL the time!!!
Even if someone over there does get miffed, so what? One minute and thirty seconds of them clucking over by the coffee machine is worth N-O-T-H-I-N-G compared to the benefit that would accrue both to yourself, and the student who will take your place. At your age, your sensitivity to social pressure is higher than it will be when you're older. Take my advice - learn to disappoint people when you have to. If you don't, you'll be lead around all your life. Trust me, if you go to a college you don't want to, you're going to look back in ten years and ask, "Why the hell did I give up on the better college - just to save some theoretical person from theoretically disapproving of me? I don't owe it to anyone, to get a lesser education than I can get, just so they don't have to be irritated for a minute. Saving them from that meaningless irritation has cost me so much!!!"
You don't HAVE to make that mistake, so DON'T.

I agree with your general point, but you seem to have totally elided the difference between college and grad school here.  Grad school applications are read by the professors you would be working with in the program, and incoming classes are often in the single digits (depending on the department).  It a MUCH bigger deal to accept and then back out of a graduate admissions offer than a undergraduate offer (which I agree you should have absolutely no compunction about changing your mind over).

In particular, the people you annoy and inconvenience are not admissions officers, they are senior people in your field who you might cross paths with in the future.  I agree that the benefit to the student (especially of a funded vs. unfunded offer) is big enough that it's still a reasonable decision, but it's not as trivial as you make it sound.

Ah, you're right. I missed the distinction. I've been awake for too long! I will amend my post.

@Huns: for future reference: if the question had been about undergraduate admission, it would have been off-topic for this site.

Thanks for letting me know. I had no idea until you said so. It would be nice if they hinted at that in the header somewhere. By all the topics that get put on hold, it seems like mine is a common mistake.

It is likely not a legal issue unless there is some sort of contractual obligation.
However, to be polite and professional you may want to consult with the place you accepted first.
Many universities want whats best for their students, and will advise you based on what they can offer you.
If it hasn't been too long, you shouldn't be putting anyone in a pinch.


text:
Why do many graduate programs in mathematics (United States) still have foreign language requirements today (in 2010s)?
As far as I know, nowadays most of the mathematical literature is written and published in English and mathematicians communicate with each other in English. Although there are certain number of books written in other languages (like EGA), but at the same time their counterparts also appear in English (like Stacks Project). However, many graduate programs still require their students to pass a language translation test in French, German or Russian (which a paper dictionary, not a dictionary app in cellphones, which seems even more ridiculous to me...). I wonder what makes it still necessary to have foreign language requirement as of 2010s.
I believe my question has different focus than this one
Mathematics Ph.D program foreign language requirement
Where the questioner specifically asked for advice for the most useful language among French, German and Russian:
I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career.
while I am asking why we ever need a second language for mathematical study in 2010s:

Are those programmes taught in English?

It can be quite helpful to know another language, as is nicely evidenced in this answer to a related question.

@fqq Sorry I forgot to add "United States"

Good luck reading the SGA if you only speak English. The proficiency required is not really that much, just basic reading comprehension.

One thing about mathematics, compared to other technical fields, is that older literature doesn't tend to become obsolete.  So although most new mathematics is written in English, you will often still need to refer to older books and papers, written before English became dominant and hence likely to be in other languages.  This may not happen so much in, say, computer science.

As a matter of fact, there's a paper in French that would be good for my advisee to read, but since he doesn't know French, and it's just one paper...

It seems that some programs are doing away with this, recognizing the lack of importance. Mine just got rid of the requirement last year.

Just an anecdotal answer: My mathematics PhD program (at UC Berkeley) had a requirement to pass a language exam. This involved translating to English 1-2 pages excerpted from a mathematics paper written in French, German, or Russian (student's choice). We were given several hours to do this, and the use of a paper dictionary.
I did not, and still do not, speak a word of any of these three languages (unfortunately). But I took the exam in French and passed easily thanks to the high number of cognates between mathematical French and English, context clues (being somewhat familiar with the subfield of math helps), and the fact that I had plenty of time to look up any words I didn't know in the dictionary.
The main thing I gained from this experience was confidence that with a little extra effort, I could actually read math written in French. Since then, I have at times had reason to use this skill that I didn't know I had.
I believe Berkeley math did away with its language exam requirement the year I graduated (2016), and I think this was an entirely sensible thing to do. But I would encourage all grad students to get some experience reading papers written in a language they don't speak!

My experience at UC San Diego was similar, except that it was 5-6 pages from a book, and I got a week to translate it, using any available dictionaries or translation tools.  A classmate had gotten somewhere a "mathematical French" dictionary, about a dozen photocopied pages, that focused on words like "therefore" and "conversely" and so on; it was extremely helpful.

It might be worth mentioning that there are plenty of important math papers (even today!) written in French, let alone the super important historical papers like Deligne's Weil I, II, Hodge I, II, III, SGA,  etc.

This was similar to my experience in the same program (finished 1989). As it turned out, my dissertation required reading one paper in French (easy) and two in German, one by Hasse so difficult a native speaker told me it was almost incomprehensible.

Indeed, this is exactly right about at least one aspect of reading mathematics not written in one's first language: it is in fact possible to arrange to understand them, although obviously more slowly. That is, more doors are open than one might have imagined...

This is the same as how it works at Chicago, and I’m glad it’s still required. I believe the language list is the same as what you’ve mentioned, though there’s discussion of adding Chinese

I noticed something very similar watching somebody with mediocre English skills teach physics. The math-heavy parts were comprehensible because he could write equations and say "derivative" or "substitute" and wave, but when it came to the conceptual physics parts, everybody got lost.

Same here at Harvard in late 1990s.  My roommate was a religious studies grad student who knew lots of languages.  The weekend before the exam he and I translated a five-page paper from French using his language knowledge and my math knowledge.  We were like two blind men feeling the elephant.  But I learned enough to pass the exam.  Tangent: Two years later I fell in love with a Québécoise and had to learn another kind of French altogether.

Part of it is just inertia. But it is a small part, I think. When I studied maths in the previous century there was a two language requirement. Initially it was French and German. Russian was added as a third option when it was realized that a lot of great math was being created in the USSR that wasn't available in English (or French or German). Later, one could substitute a Programming Language for one of your two languages.
However, even today, not everything that a working mathematician wants to know is available in English, so for a practical reason it is useful, still, to have language skills beyond English. Machine Translation has made great strides in the past decades, but mathematics is probably still very difficult to translate correctly. This is partly due to the smaller sample size of available texts on which to train translators.
But, I would, myself, be hesitant to drop a language requirement from a modern mathematics graduate program for a completely different reason. Consider, as I do, that language skill is a help in mathematics itself. Among other things, mathematics is a language, and it requires a certain training of the mind in order to speak it well. Mathematics uses vocabulary and structure to express deep ideas - language. So, language training of any sort, trains the brain in a certain way that may actually assist in the mathematical way of thinking.
I'm not sure I'd be adamant with my colleagues who wanted to replace the last language requirement in a program with something else, but I'd want to hear arguments about how that would make for better thinking. Another math course or two might be useful, but would it be better? Hard to say.

“Mathematics uses vocabulary and structure to express deep ideas - language. So, language training of any sort, trains the brain in a certain way that may actually assist in the mathematical way of thinking.” This is absolutely true, but it would seem that just more training in the mathematical language would help more. So far as I am aware, studies have shown that the best training for any activity is that activity; related activities may help, but not as much. Food for thought, anyway.

@KRyan: I'd hesitate to guess that it'd help in a different way, though. Maybe practicing A helps more with A than B does, but B might help in ways that A doesn't. And if you're going to do A anyway, then you'd get the practice you need from A, whereas you might not get what you'd need from B...

For language exams at math departments I've seen, they require such a rudimentary familiarity with the language that I don't think preparing for them could be considered "language training."

@JonathanReez the brain isn't about "space". It is about organization.

In Italy there is a foreign language requirement for engineering faculties. English language obviously. But I can tell you with a decent amount of confidence that even if you pass that exam, you still can't read/write/speak/understand English anywhere near a decent level, let alone understand mathematical texts, just because of that exam. You have to put the effort into improving beyond the requirement if you want it to be useful. So... Is another language useful? Absolutely. Is the exam itself useful? Not so sure.

@ChatterOne It is probably a compromise between what they want (all students to get an usable English skill) and what is reachable (to lose only a few students because they can't learn it on that level).

As a thought experiment, let us assume a hypothetical scenario in which there was a tradition for mathematics graduate programs to require their graduate students to take a cooking class. Someone would then come to academia.se and ask: why do they have this requirement? Several well-meaning, well-intentioned, extremely intelligent people would then post answers offering quite rational explanations, which might go along the following lines:

Why do mathematics graduate programs require students to take cooking classes?

Mathematics is an intense activity that requires a lot of energy. Food provides energy. Hence, a mathematician who can cook well and efficiently will be more productive than one who can’t. Moreover, a mathematician who can make tasty food will be happier and free from the distraction of constantly thinking where to find tasty food. Again, they will end up being more productive and producing more and better mathematics.
Mathematics is a social activity. If you can cook well, guess what? Lots of really good mathematicians will want to be your friends and collaborate with you. You’ll produce more and better research.
Mathematics is a language, and cooking is also like a language. Learning how to read and execute a recipe, which is really an algorithm written in a kind of pseudocode with a particular grammar and syntax, is a skill that will carry over well to many areas of mathematics (combinatorics, logic, theoretical computer science, and much more).
Cooking develops your brain. Cooking requires a substantially greater intellectual effort than ordering food at a restaurant. Naturally, having a better developed brain will make you a better mathematician.
Cooking teaches you to care about the order of operations. Ever tried making a recipe and added a vital ingredient at the wrong stage, with disastrous results? I bet you’ll never put your operators in the wrong order in your next algebra paper!

Etc etc.
To summarize, the tradition of cooking classes in graduate programs is 100% logical, and should continue. It is not at all due to inertia. I’m sure graduate programs evaluate their requirements all the time, and are constantly considering the benefit provided by any requirement against the opportunity cost of not replacing that requirement by something else.

Also, since cooking is required by most programs, a program that drops that requirement risks damage to its reputation.

A language is a general purpose tool that can be used to use other tools. A meta-tool if you want. Cooking is a tool in itself, not without its own benefits, but definitely not comparable to a language. You might even need another language to understand recipes. So, I don't think this is a fair comparison. (And just to nitpick, from a linguistic point of view mathematics is not a language, because there are no native speakers)

@ChatterOne The requirement that  language have native speakers seems bizarre to me. Many constructed languages have no native speakers - does this mean they aren't languages? If I raise my child to speak Klingon from birth, does Klingon suddenly become a language, when it wasn't before? Does a language lose that status when its last native speaker dies? I'm curious if you have a reference showing that this is a common view.

@AlexKruckman This was exactly the argument that CS PhD programs used to get rid of their foreign language requirements. "Our students already speak C like natives; why do they need to learn French?"

@JeffE Things like C and "mathematics" certainly have a different status than natural languages. I'm not necessarily disagreeing with ChatterOne's assertion that "from a linguistic point of view mathematics is not a language" - it depends what exactly what you mean by a language. My point was just that I think it's bizarre to take "no native speakers" to be the distinguishing factor!

@ChatterOne your description of a “meta-tool” (which I rather like) as a tool that can be used to use other tools also fits mathematics itself. Yet for some reason I don’t see French language grad programs requiring their students to master the meta-tool that is mathematics...

(Remark: This answer was written while it was not clear that the question referred to graduate programs in the US only.)
Some graduate programs may come with additional requirements, e.g. teaching duties in the local language or come with a working contract and this may require a certain visa and the visa requirements are not controlled by the graduate programs.
Also, living in a foreign country is much easier if you have basic knowledge of the local language, e.g. for communicating with landlords, offices, or the university administration. And no, English is not always enough (e.g. in France or Germany, you will encounter situations where English is not very helpful).

These are valid considerations, but I don't think the language exam addresses either; such exams typically focus only on one's ability to read mathematics in the target language(s).  This is next to useless for any sort of non-mathematical communication.  Even the grammar one learns is quite limited.

Learning a language is quite useful generally. However, based on this answer, it seems that PhD programs in _________ (fill in blank with any subject) should require second language requirements. After all, it could be quite useful to be a botanist and know Portuguese if living in Brazil. It could also be useful to know about local foods and traditions. But none of this is required knowledge to get a PhD in botany.

I heard a story about a mathematician who tried to extend mathematical language knowledge to general communication, and as a result would say things like "Let us consider a bottle of wine".

@NateEldredge: I've known mathematicians to talk that way in their native languages.

Not all math papers were written in the 21st century, and not all have been translated into English.
When I did a Math research paper in college, I found that all the books and papers I needed were written in the late 1800's, in French.  Many were not translated into English, and most of the ones that had been were lent out.
My professor was mildly amused to see that my bibliography contained only French original sources.  (I finally got some mileage out of AP French.)
The era in which the work you're interested in was done will dictate whether it was most likely written in French, German, Russian, or English.

It sounds like you did more of a math history project than graduate level research in math, no? Perhaps that isn't the best reason for a graduate program to make a foreign language mandatory for all their students.

@Anyon, I think your comment/question does accurately reflect the viewpoint of many grad students... but not so many faculty, insofar as the state of mathematics in the late 19th century was very close to our contemporary viewpoint (quite in contrast to other sciences, apparently). If I can have research students who have really assimilated the research-level mathematics prior to WWII (up to which time only a tiny fraction of serious mathematics was written in English) I'd be wildly happy. :)

@paulgarrett I'm probably just biased by my non-maths perspective, but it certainly seems unusual to me that all the references would be that old.

@Anyon, maybe not all, but it is conceivable that many would be. Many serious, and still-not-resolved, issues were raised then. The only reason we don't see more references to those days is that very many people are not aware... apparently thinking that nothing worthwhile happened before they were born, etc. :)

I venture to suggest that Latin should be in the list of languages in the last sentence.

While I can mostly read Latin (from 7th & 8th grade), I've never had to use that facility for mathematics that is still considered relevant.  Maybe for History of Mathematics it could be useful, but I only wrote 1 paper for a single undergrad course in that, so I wouldn't know.  But seriously, the Romans didn't even have zero.  (That came from India by way of the Arab world.)

To answer the literal question at the end of the post: yes, I think the optimal second language after English is French, ... and my own students in number theory, automorphic forms, representation theory, etc, would be disadvantaged at not being able to read Sem Bourb and many other things. Yes, the "classic" number theory stuff (Hecke, especially, but also to some degree Siegel) has been rewritten in English... but not everything, and as expected with some "lossiness".
So, sure, one can seemingly "survive" English-only, but I'd feel awfully claustrophobic if I had no idea what those dang not-English-writing French people were doing these days, ... not to mention Germans. Perhaps ironically, it appears that most Russian writing has an immediate English equivalent. (Not at all the case some decades ago.)
It's a question of how to best spend one's time, sure.

The reason why mathematics programs ask you for at least one foreign language is not necessarily so that you can read or translate older mathematics papers.
The reason is epistemological as how we come to know of objects, their organization, processes and future dynamics.
Mathematics contains one particular view of the world, which can change and evolve according to the individual's sense of perception.  This evolution is aided by learning a language different from the individual's original language which has painted, constructed, and framed its sense of reality.
Learning a different language, particularly that of significantly different roots from your mother tounge has the potential to open you up to a new set of perceptions, of grouping or organizations of primitive mathematical objects.
Thus it expands your epistemological reference point of view and can potentially open you up to the creation or modifications of new mathematics.


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Job market in academia: applied/computational math vs. pure math
Many people told me it is much more easier for an applied/computational mathematician to find a job in U.S. than a pure mathematician. I believe that it is largely due to the fact that applied/computational mathematicians have many advantages in finding jobs in industry. I wonder if this is still the case in academia, in particular research-oriented schools. Well, at least in the department I am studying at, most postdocs and newly-hired assistant professors are doing applied\computational math or potentially applicable math.
Specifically, Is it generally easier to find a job in research-oriented schools (regardless of ranking) if one works in applied math rather than pure math? (I naively believe that there is no special advantages for applied mathematicians to apply for a job in teaching-oriented schools. Please let me know if I am not right)
To avoid the tag that "this question strongly depends on individual factors", I want to clarify the words "generally easier". I want to know whether or not most research-oriented math departments (specifically refer to those with both pure and applied mathematicians. I know in some universities, pure math and applied math belong to different departments) are more interested in hiring applied/computational mathematicians as postdocs/APs and the reasons behind it.
Answers based on personal experience and objective evidence/statistics are both welcome here!

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

You can answer your question yourself with a bit of work. The American Mathematical Society has a job page for recent Ph.D.s. You can use it for free.
From the inside of a (now disjointed) Math and Computer Science Department, it looks like Deans are much more interested in applied mathematics as they hope for attracting outside funding and NSF grants. But this is just anecdotal information.

Choose the odd one out: a pizza delivery guy, an engineer, a Walmart cashier and theoretical mathematician. The answer is theoretical mathematician because the other three can feed a family of three.

Granted the above is just a joke but whenever you've got a choice between theoretical and applied math, the choice should be the latter at least from the perspective of career benefits.
Now to be specific to your question: I would say that, all things being equal (which they rarely are), applied math still retains its job advantage over theory even in academia. Both aren't resource-intensive: all you need is pen, paper and a decent computer to do research. However, there will probably be more funded projects available and more opportunities for inter-disciplinary research in applied math. Companies and institutions really want to be able to model and simulate complex phenomena and are eager to sponsor and collaborate on that sort of research.
Specifically, in my field (reacting flows), mathematical modeling and accurate simulations save a lot of money and time that would have otherwise been wasted building prototypes and conducting numerous trial-and-error experiments. So, people in the field collaborate with applied physics and math folks all the time. |
Disclaimer: My field is not math and I'm hazarding a guess.


text:
Can we use online video lectures to free university professors from lecturing for lower-division courses in mathematics?
During the 2020-2021 Covid-19 season, universities and colleges around the world made most of their math classes online over Zoom and many professors have their recorded lectures.
Most math lectures usually don't involve in-class discussions and for the most of the time, professors dominates the teaching. Oftentimes, excellent recorded lectures can be much better than live lectures in mathematics and other theoretical sciences. I have seen many students commenting under certain YouTube videos that those recorded lectures are much better than their professors'.
I wonder if we could possibly use those recorded videos (there are a lot of such videos modulo-necessary-editing during Covid-19 season. Building a system like (https://nptel.ac.in/) should be feasible) to free university math professors from repeatedly giving lectures on lower-division math classes like calculus, linear algebra or differential equations (but at the same time, asking them to hold discussion sessions and office hours to have more interactions with students and of course, grading homework assignments and exams), so that they could have more time on their research and teaching classes that are  more relevant to their research? I believe this is particularly important for the early-career mathematicians. Many postdocs in math are assigned with lower division classes that are kind-of digressions to their research.
This seems to be a win-win strategy for both professors and students in terms of the mathematics. If this is good then maybe it should have been conducted already like https://nptel.ac.in/. What would be the potential challenges for doing so for higher education in United States and Europe?

I think this would most likely lead to greatly reduced sizes of math departments... Automated by playing those videos over and over. And, yes, many students might rightly be happier with good videos in comparison to dubious live lectures...

Can we? Yes. Should we? Harder to say. But this looks like a call for opinion only. Hard to say how less interaction between students and professors can be called a win for anyone.

@paulgarrett Well, math departments are funded by tax payers after all. The government should know what to do, I guess? We need mathematicians not just for their teaching.

@Buffy I hope the last sentence in my post would make this question less opinion-based

@NoOne, I agree that "teaching calculus" is not the only function of mathematicians, but from an all-too-common viewpoint of university administrations, a way to reduce costs of "covering" calculus would be highly desirable. The other parts are toooo intangible. :)

And your last paragraph starts with an opinion. Before you think about "how" to do so, settle the question of "why".  Color me skeptical. No single explanation of a concept will "reach" every student. Flexibility is needed. Replaying a video with the same explanation repeated will be sub-optimal. Some will be fine. Others not. One thing beginning teachers need to learn is that answering a question with exactly the same explanation is not going to be successful. That is why the question was asked.

@Buffy Well, this is a question that I want to ask for years but haven't thought of a way to perfectly avoid the "opinion-based" tag.

@Buffy "Replaying a video with the same explanation repeated will be sub-optimal". This can be address by professors holding discussion sessions and offices hours. If they are free from lecturing, they would have more time and energy for that.

@Buffy "One thing beginning teachers need to learn is that answering a question with exactly the same explanation is not going to be successful" I believe nowadays new teacher, aka postdocs, need more time for their research just in order to SURVIVE in academia... If there are already nice video lectures recorded by experienced teachers, why do we still need to ask young mathematicians' to spend so much time on teaching?

But consider the time delay between the time the question arises and the first opportunity to get it answered. And it would be more chaotic, with only the one asking the question hearing the answer. Often if one student has a question, others have it also, but aren't "brave" enough to ask. Not a good plan.

Among other things, teaching gives you additional insight into your subject. Efficiency and effectiveness are not friends here.

@Buffy "But consider the time delay between the time the question arises and the first opportunity to get it answered". (You comments seem to make it more necessary for me to add explanations on "how".) Yes, this is an issue. But recored videos can be edited. If a student want to see some certain details about L-Hospital's rule or more clarifications on Jordan normal form, professor can add explanations to the original video through editing.

@Buffy "Among other things, teaching gives you additional insight into your subject. Efficiency and effectiveness are not friends here." Yes, but as an algebraic geometer or number theorist , a person may want to spend more time teaching algebraic geometry, algebraic number theory or at least abstract algebra, rather than calculus and linear algebra...

@NoOne, universities need far fewer people giving courses on higher mathematics than on calculus. You want math depts of 5 faculty + thousands of hours of videos?

@paulgarrett Then it would be the government's duty to provide math departments enough grant to maintain enough number of mathematicians... Providing adequate funding to fundamental research should be in the radar of lawmakers...

"Then it would be the government's duty..." Ha. Good luck with that.


text:
How do I balance interests within a specialized PhD? Do I simply read a lot outside my field?
I'm going to get a Master's in Artificial Intelligence, likely a PhD in Molecular Biology afterwards. I'm thinking I can do computer vision as a hobby while putting most of my energy toward understanding the big picture of cellular processes, because computer vision is something I can performance check. That's why I'm leaning towards Molecular Biology over AI even though I'll have years of research experience in AI.
The problem is that I want to know too much. I want to study Cell Signaling/regulation, Genomics, Molecular Biology/Biochemistry/Structural Biology, and disease during my PhD to do some interdisciplinary research in all of it. I want to develop computational data mining models that can take entire cellular systems into account.
Is it unheard of to fully specialize in two subfields during a PhD and associate my research topic to cancer later on in my PhD over 5-7 years? Or must I, for example, study a single new enzyme in an important pathway across various environments only in biochemistry?
Can you have a more general PhD that is still a successful contribution to the field? Say an analysis that studies an entire biochemical pathway from genetics through molecular biology to cancer if I have two committee members (one in each subfield)?

One step at a time. At this point of your career path, very early, focus is important in order for you simply to get to the next step. Many of us tend to have all sorts of things that attract us, so nothing unusual there, but prioritization is what saves us.

You say, "The problem is that I want to know too much." You don't want to know too
much, but maybe you want to know too much all at once.
If you are going to pursue a PhD in molecular biology, you owe it to
yourself, your department, your research adviser, and whomever is
providing financial support to focus you energy on learning that
material and writing a fine thesis roughly within the usual time
expected by your department. Most serious PhD students don't have
much time for 'hobbies' outside of their coursework and research,
but if you do, you could consider AI as your hobby for now.
There seem to be unexploited connections between molecular biology and AI,
so it is worthwhile to look forward to exploring those connections
at an appropriate time.
If you can't bear to let a major connection to AI wait until you
finish a PhD in molecular biology, then maybe you should re-think the
order of your concentrations. Both fields are developing rapidly.
Will it be easier to catch up on recent developments in AI after
your PhD in molecular biology? Or would it be easier to learn about
recent developments in molecular biology after getting a PhD in AI?
I have some idea how much biology an AI a person would have to know in order to
make worthwhile applications to molecular biology. But personally, I
have no idea how much you would have to know about AI
to exploit opportunities in molecular biology. So I have no advice about the best
order of concentration.
As a starting place for you to make the best decision on your own, do you see how study "of a single new enzyme ..." would make successful exploitation of AI techniques more likely? Or maybe you can find a molecular biology department
that is already deeply committed to using AI.
I have done a lot of work in applications of statistics and probability modeling
to biology with only self-taught biology (not even a high school biology class).
My impression is that it is easier for me to catch the essence of the necessary
biology ad hoc than it is for biologists to catch the the essence of the
required statistics or probability. (Some of my collaborators may disagree
on that.)

Thank you so much for investing the time into your response. I'll be reading it a few times over as I continue to consider everything you have said.


text:
What are the duties of the reviewer when reviewing a literature survey/review paper?
What are the minor and major concerns that the reviewer should be concerned about and what improvement suggestions should they take in mind when reviewing a survey/review paper?
Are their duties for checking the language of the writing? Are they to check for the figures are in sufficient resolution?

reviewer is the person who is responsible for reviewing the paper or the surveys that will be accepted in the journal. Does this the appropriate place for this question?

it is a review paper that serves to survey/review an area of literature

https://academia.stackexchange.com/q/148027/13240

If you are new to this, ask the editor of the journal for guidance. They probably have some prepared.
The minimum is to make a judgement about whether you think the paper should be published at all. If it is obviously sloppy or incorrect then you don't really need to do more than say that you think it can't be salvaged.
But, to serve both the journal and the authors you can, for reasonable papers, provide any advice that you think will (a) be required to make the paper come up to the journal's standards and (b) improve it for readers.
For a review of the literature you can point out that parts are missing if you notice that and possibly provide some pointers. But your job isn't to complete the paper.
This can include some minor suggestions on the writing and on the quality of the images. But you aren't expected to be a copy-editor. But if the language is terrible, you need to say that and, if possible, say how the author might go about getting it improved.
There are very rare cases, say in mathematics, in which the paper contains a poorly written but very important result. Then you, the editor, and the authors have the possibility of getting something important out, even if it requires a lot of work. For such things you can make more extensive suggestions if you have the time.

thanks for answering. but if the survey is a too-short search like 7 studies on the specific point but there is a lot of it .. what should the reviewer comment on that? and can the reviewer ask to add a new section to make a survey better like add a table for the previous studies on the point of the survey with its advantages and disadvantages ? as an abstraction

Anything you notice that you think will improve the paper can be mentioned. Mentioning it early will make later decisions easier and ease the work overall. Don't hold back. You aren't the only reviewer in most cases.

great thanks for you

Absent some compelling reason to the contrary, I would think that the reviewer should review every aspect of the paper that is within their ability to review.  Check the content, the writing, the scope, the figures, and anything else you have the ability to check.  If there is some aspect of the paper that you don't have the ability to check (e.g., it is out of your field of expertise) then it is useful to state this limitation in your review.  There are some aspects of papers that are also checked by copy editors (e.g., spelling, resolution of figures, citations, etc.), but it doesn't hurt to raise these issues in the initial review process, so that the revised paper fixes them before going to the copy editors for production.
For a review paper, the main contribution you can make as a reviewer is to point out any deficiencies in the scope of the review, or the summary of papers in the field (e.g., identifying additional papers that should be cited, or places where the paper is summarised incorrectly).  However, there is no reason that you should not point out any deficiencies in the paper that you think could be improved.  Most reviewers try to respect the "style" of the author, but if you have suggestions for improvement then I see no reason to withhold them, even if they are minor.  Bear in mind that in the revision process the authors may choose to act on each of your suggestions or decline to do so, so you are not forcing anything on them.

Thanks for your answer


text:
How to appropriately email a researcher to ask for details about their paper?
Often when I am reading a newly published article, I will encounter some points that are difficult to understand. For example, some details of the methodology that I think the authors did not explain in enough detail.
I'm not sure how to write an appropriate email to ask for clarification, especially on how to phrase, so as to be polite. For example, should I write "Ask for details on methodology" or "Inquiry on details on methodology" or ...?

You can also ask if there is an appendix (published or not) containing more methodological details than appeared in the published article.

Does this answer your question? How should I phrase an important question that I need to ask a professor?

Be warned that authors of papers can be of varying degrees of helpfulness when you ask them about details of their papers or more explicit details of their methodology/computations.

There is a website called pubpeer, which is useful for public comments of papers, if you have a question it's possible other people may too, maybe someone has asked it before

By the way, it should go without saying but also make sure you have thoroughly read the paper before asking :)

Thanking them for their work at the start is a good idea. Learnt this habit from my PhD advisor.

I would not care all too much about being polite. Explain what you think about the article, as in, give a usefull feedback, then tell you did not understand XXX , or maybe that the author's assumption their readers knew about this topic was not accurate, and ask about some insight. Oh, and it always helps to ask for sources, not direct information. I assume that you already did a complete search for the topic, and if you did, you should note that too. Showing effort is worth more than being polite

Showing interest is nice, accusing people is not. So, show interest, something like "I saw your article XXX in YYY and I am working on something similar. I was wondering how exactly you dealt with ZZZ."

This probably is a bit culturally determined, but almost anything polite will do. I personally prefer your second formulation, but others might not. Don't overthink it.
But in a first mail to the author(s) I suggest asking questions that can be answered fairly simply/quickly and don't ask for too much. If it seems like it will be a lot of work to reply, then you might not get any answer. You can always follow up with additional questions if the authors seem open to communication.
In the best case it can open the possibility of future collaboration if the topic is of mutual interest.

Contacting researchers for clarification is good practice. Before you do, ask yourself why you think the authors did not provide enough details. In formulating an answer, you may find they did provide enough details. Otherwise, you'll have established a better understanding for what you need to know, and you can put that to the researchers. Always be humble in asking.

Let's get more templated. What about this?

To: %Corresponding author% Subject: %Paper title%
Dear Prof. %NAME%,
With a great interest I read your paper on %TOPIC%. Since I work in
a similar area, I would like to %short presentation of what you want% [e.g., to compare my approach [1, 2, 3] to
yours].
Do you have %your actual inquiry, detailed%?
Thank you very much in advance.
[1] Paper [2] Paper [3] Paper
Best regards,
%Your name%
--
%Your signature, including your institution, email, phone, and further ways to contact you%

If you are asking for code, try searching GitHub and further usual places first. Googling the corresponding author would also help. Take a look at their most recent papers, may be your question is already answered.

What if I'm not sure if they are a professor or not.

Just google them?

So long as what you write is not rude or arrogant, nobody will care much.
The important thing is to ask some specific questions. If you just asked me for "details of methodology" I would hit the delete button, because I'm not going to write a comprehensive reply that is probably longer than the published paper telling you every little detail about what I did - especially if the paper was published years ago and I have forgotten most of the details anyway.

That depends imho. If you write an article about Nuclear Power and completely leave out the part about uranium atoms splitting, well, do you assume your readers will know ?

@clockw0rk yes? For a journal article, I think you can safely assume a decent general science background, especially in the topic of the journal (e.g., fission for a Modern Physics Letters paper).

@Matt u maybe right about that, but I encountered papers about advanced hacking techniques where the author completely leaves out facts about what he is refferring to when he says "... because it is simply a fact" or "...as is common knowledge". Not the topic of this question, but definately provide at least sources to your readers where they can find the basics of your research. Well, I guess it's part of the topic that these papers always come a little "mysterious" or "arcane", so to speak.

I was for 5 or so years a university researcher and co-wrote a few papers. I would have been thrilled if someone had written to me asking a sensible question (they never did) but horrified if they had found an error.
Professors, on the other hand, can be time poor, so it will be best to write to the most junior author if this is an option.
In my experience many academics build upon their previous work and answers to any questions will often be found in previous papers.
To be specific I would suggest: "Do you use the [your best guess at the techniques/methodology used] system in your research?" after a preamble much as Oleg has given above.

A while ago I also read a paper and had a question which was not answered in the paper (or maybe it was and I just wasn't capable of interpreting it).
I searched for the address of the main author and wrote a polite, kind email without much fluff (because I had often heard that scientists don't like fluff).
I also mentioned that I had asked another scientist I knew first, but that no one knew the answer (just to add some justification for why I was writing him instead of asking others or consulting books).
For me, he was something like a famous Professor and I thought he would never answer my mail. But not even a day later I received a kind and helpful answer from him.
I would also like to add here to encourage everyone to answer emails like this if possible. For him it was maybe just an email but his answer meant so very much to me, I nearly even cried because I was so happy and I felt taken seriously and acknowledged.

If the paper is published it is because the editor (and presumably the referees) believe there are enough details either in the manuscript per se, in the references or that the procedure is sufficiently well-known not to waste time on it.
Thus I would encourage you to be very careful in suggesting there is not enough information: it might not be enough information for you but presumably it’s enough information for that typical reader of the journal.
You might ask for clarifications on a few specific points but do so selectively, making sure you include significant context and references so that your query is legitimate.

This is what I used to do when I was a math grad student:

Be extra polite (I would start with "Dear Prof. NNN")
Introduce yourself. Say what level you're at, what institution you're at, and who you're working with. (Briefly.) It will help them understand what level you're at and indicate a connection.
Be somewhat detailed about what you do understand. Don't just say "I didn't understand this step in the methods." Say "When you say that you did XXX, did you mean YYY, or ZZZ? Or perhaps I misunderstand completely?" Just like with Stack Exchange you want to make it clear you put some effort in, and you also want to make it clear just what needs to be explained.
Include a phone number in case they want to talk further.

I want to re-emphasize the importance of being clear about where you are uncertain. You want to make it as easy as possible for them to answer your question. And as you try to put your confusion into words, it may actually become more clear to you.
But don't be afraid to make the contact. It could even lead to a future collaboration. Connecting with other researchers is a good thing to do.
One other note: Even though I suggested "Dear Prof. NNN", that's just for the first contact. After that, look at how they talk to you. If they close with just their first name, that's generally an invitation to address them that way.


text:
WCAM speaker invitation: scam, and how does it work?
I usually receive many emails from scam journals asking me to publish my work there. And as I understand the scam is that they charge you a lot of money for that, and/or they steal your authors' rights.
But I recently got a different email (and a reminder) from Ms. Elsa Wang about a conference 9th Annual World Congress of Advanced Materials 2022 (WCAM-2022) in Tokyo, where I'm supposed to be an invited speaker/chair:

On behalf of committee and based upon your outstanding contribution in
this field, we cordially welcome you to deliver a presentation as
Speaker/Chair regarding your recent work [...]

The rest of the email is pretty well written, and the website claims to have important speakers who I recognize and have a pretty good CV. I guess this email is a kind of scam, but it's not clear to me how the scam would go.
So do you confirm this email is a scam? If so, where is the trap? (I guess all expenses are covered for invited speakers usually, no?)

Not every reputable conference covers all expenses for their invited speakers - some will just not have the budget for this. And in a scam, bait-and-switch tactics are common.

Possibly related: https://academia.stackexchange.com/questions/9362/how-can-i-determine-whether-a-conference-is-reputable

I  deleted the reference to a specific person - I'm not sure we should be doing that.

The key point is how much registration you still have to pay.  Unless the email explicitly states you will have to pay nothing, or states the discounted registration fee, it’s likely a vanity trap where you will have to pay registration and everything else (travel, hotel etc).  This kind of invitation is generic.

You'll have to pay money to register for something, it won't be free to attend.

I got a similar email from this conference. It asked me to speak about RF superconductivity. Not my field. Definitely scammy if you ask me.

A good technique to research questions of the form "is X a scam?" is to google "X scam".
I just googled "Annual World Congress of Advanced Materials WCAM scam".
One of the first hits is someone's blog, which describes the author's experience:

I’m answering a spam email from BIT Life Sciences, which inviting me
to their WCAM-2018 conference in Xiamen, China. I suppose the
conference is real (see the cover picture), but for sure it is a
predatory conference from a for-profit organization.
So, I created a fake email with a fake name and started contacting Ms.
Elsa Xing, who sent me the spam email.
...

followed by a hilarious e-mail exchange with the scammer.  TLDR: the scammer wants your money.