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How to improve student seminar presentations and seminar papers
During the last semester my supervisor Prof. X held a seminar for bachelor and master students (mostly computer scientists). According to the study regulations the students have to

give scientific presentations (based on journal or conference papers) and
hand in seminar papers on the topic of their presentation (i.e. their papers should cover the results of the original paper, lay out details of proofs, add additional explanations or examples, etc.).

The final grade consists of 2/3*presentation + 1/3*paper.
Prologue.
At the beginning of the semester Prof. X's research assistants compiled a list of interesting research papers. At the first seminar meeting the students were able to choose the paper they would like to present to the rest of the group at the end of the semester. Furthermore Prof. X made it very clear that understanding the topic is just a portion of presenting and writing down scientific ideas. Every student was assigned to a seminar supervisor to whom they could talk when they ran into problems.
Half-way through the semester there was an obligatory meeting with the supervisor and a deadline for handing in a draft of the presentation slides. At the end of the semester we organized a little "conference" and the students presented "their" papers. Afterwards, they received written feedback on their presentations from all participants (students, research assistents, and Prof. X). Six weeks later they had to hand in their seminar papers.
Problem.
Unfortunately the quality of many seminar papers is relatively poor; even the papers handed in by students who had understood their topic "quite well" and had given good presentations are surprisingly different from what we expected.
Main issues:

Even though they were allowed to use their native language, many papers had bad spelling (obviously no spell checker was used) and bad grammar.
In some cases it was impossible to understand the basic ideas if one had not already been familiar with the subject.
Imprecise language and almost no sources cited, e.g. "Algorithm Y is rather efficient in comparison with other algorithms." [citation needed].
Some students only cited a single source (= "their" research paper).

Questions.
How should we address these issues? Of course, we are going to give some feedback on their seminar papers. I am worried about the next seminar. Reading all these papers was (mentally) exhausting. Should we require that seminar papers must be handed in first? How could we install an iterative feedback process?
I am looking forward to your ideas and experiences.

For most people, it can take years to understand how to write decent papers: it seems to me that you want your students to accelerate this understanding in one semester.

I am aware of that and I don't expect a student seminar paper to meet the same standards as an accepted high-rank conference paper. I know that writing down scientific ideas such that people (who are still unfamiliar with the subject) can read and understand them. Especially this is hard for bachelor students writing their first seminar paper. But the issues 1, 2, and 4 (listed above) should not occur on a large scale.

Why not? Look at Stack Exchange questions across different sites: How many poorly structured, poorly written and unclear questions are there? How many people doesn't cite the sources of images, diagrams etc.? This happens also to native speakers and to graduate students. And a question is much shorter than a paper. For instance, for what concerns point 4, I had no concept of citation until I had started my PhD. This is the reason I stopped asking my students to write lab reports: they are generally awful and I don't have enough time and energy to improve a hundred lab reports.

The seminar papers are mandatory according to the general study regulations of our faculty. Hence, we must not skip this part of a seminar.
To be honest, I was hoping that this thread would attract more attention.

Did you require draft papers from the students and give the students suggested corrections?  Student papers will improve if you require this, particularly if you go through multiple cycles.  It's very labor intensive though.

This sounds exactly like the situation we are experiencing at our university as well. We have improved the situation by forcing the students to write papers and similar texts every semester (starting from semester 1), in different settings, and we're giving detailed feedback afterwards. In our first and second semesters we are using this setting for one course:

Students can select one topic and prepare it in a group of three (the idea is to mix people so they can give feedback within their group). We emphasize that everyone will receive a grade based on personal efforts and success, and there is no group bonus / malus.
They will present the topic in a group of three (20 min presentation + 10 min discussion)
We are giving a 3h presentation on scientific writing.
They have to give their paper outline to their supervisor after about 3 weeks. The quality of feedback depends on the effort put into the structure. If they hand in "Introduction, Main Section, Conclusion", they will receive a "correct, but good luck" ;-).
They shall hand in a first version after about 50% of the course. This version will be discussed in a personal feedback session of one hour by a scientific journalist
At ~3/4, they have to submit their "pre-final" version to a conference management system.
Everyone is now switching roles and becomes a reviewer. Every reviewer has to review two papers. The quality of reviews is part of the final grade.
After the reviews. the groups have one more week to finish the final paper submission.

As said, we are following this highly intensive approach in semesters 1 & 2. In the subsequent semesters, we are having other settings, but we force them to write every semester. It seems to bear some fruits since the quality improves over time, but it's clear that we can't fully repair what went wrong during school...

Thanks for the answer. How does the peer review work out? This is an idea that we discussed internally, but we are not sure whether there is a constructive outcome. Do students see their peers' papers and think "Wow, this is so much better than mine. I will improve my own paper!" or are they not able to see that because if they could their own papers were better?

Quite well! Many students are criticizing things they did wrong by themselves and sometimes even learn from this . For me it's a great help as well since I enforce plagiarism check in this step (without consequences for the authors at this stage). The feedback is quite helpful in general.

That sounds interesting. Based on my experience, whenever students have to give feedback to presentations, they tend to praise themselves (i.e. they defend themselves against the enemy aka the grader :-)). Hence, I would suggest not to grade the feedback itself. However, how can useful feedback be encouraged if there is no 'pressure'? Ideally students should give great feedback because because they need helpful comments on their slides and papers, too. On the other hand there are many reasons not to put too much effort into other students' papers instead of theirs.

We are using a nice trick: The review is part of the final grade. The grade depends on the quality and appropriateness of the review. This means, if they are way too positive, they will get a worse grade for their review. And I do not take the reviews into account when I'm, finally going through the submissions. So honesty pays of a) because they will get a better grade and b) their friends can improve their paper.

Here are some thoughts on the four points you mentioned:
Point 1. From my personal experience general spelling and grammar skills seem to reduce, but are most of the time a matter of "I don't care enough" or probably "I did not have the time to check".
As you already did a mandatory meeting with them half-semester for the presentation you may additionally request that they have e.g. about a page of their paper written with citations, etc. Then you can already point out some details ("there are many spelling errors, please use a spell checker" or "you do not give any references, where does statement X come from?").
Points 2./3./4. If the students have never written scientific paper (or something similar) before it will be hard. I second the answer by OBu in the sense that repetition and practice is key.
Additionally, the paper they have read might be the first paper so they might not know how to cite and that non-trivial statements have to be either proved or cited.

The idea of everyone having a seminal supervisor they can talk to is great but it assumes that people will actually talk to them.
Many might think that their problem (e.g. how to cite) is too trivial to bother the supervisor.
In general the things you can do are:

In the beginning give a lecture/talk about scientific working and writing. Show how to cite properly, that citing is necessary, how to find additional resources etc.
Make additional rounds of reviews, be it either between students themselves or by having the students hand it non-final versions of their paper to the supervisor and getting feedback on them. This also helps with the possibility of people just starting to write the paper shortly before the deadline.

Thanks for the answer. Having a designated supervisor and not talking to them seems like a key issue here. Setting up an iterative framework of meetings, feedback, beta versions of slides and papers might be a good idea.

Firstly, I think courses such as yours are a really useful opportunity for graduate students to practice academic writing and presenting. I did one with a similar marking structure during my masters degree and I found it a valuable experience. Here are some further thoughts on the points you are concerned about.
On point 1:

It might be best for students to write in the language of the paper they have read. This would be helpful in situations where for example, field-specific words do not translate naturally into the native tongue and could instead be transferred from the paper. Of course, this might also worsen the quality of the written output.
In my own experience, the act of preparing a lengthy presentation on a research paper for the first time was stressful and time consuming (especially as it was slightly outside of my field of experience). I used my seminal paper as a means to document my understanding as I went along and it was never my primary focus. If you want students to focus more on the written component, you could shift the grade break-down to put more emphasis on this part which will probably naturally lead to higher quality (at the expense of the oral component?).

On point 2:

This is a hard skill to learn, especially if the research paper is not closely related to an area the student is already familiar with. Informally talking about it to their peers might be a good way to realize when they are not explaining themselves clearly. This could be done in small groups towards the end of the course, with the TA(s) moving around and providing advice.

On points 1-4:

To some extent, this is probably achievable by having a very clear rubric for assessing the written work that stresses the importance of clear work with proper citations and use of multiple resources.
Another option would be to have a short exercise at the beginning of the course (counting maybe 5%) where the students are given 2-3 short passages and asked to provide a summary, complete with citations. This could be marked fairly strictly, providing an opportunity to highlight issues.
A third option would be to do something like a "journal club" where students distribute their paper to their peers before their presentation, and one component of the presentation involves a discussion of the work where issues are highlighted. The paper could then be revised before the final submission, allowing students to concentrate on and implement corrections that they have heard over the course of the presentations (although depending on the academic culture, this might be viewed as too embarrassing for the students).

Since these courses (seminars) are mandatory, we can't take credit for that. ;-)
I agree that writing down in order to increase understanding is very helpful. If I cannot verbalize subject XYZ, I have not really understood XYZ yet. However, as you point out, this is time consuming.

It sounds like the oral presentations weren't so much the problem as the written work.  Do you want to separate out the writing goal and work on that in a separate course?  Or do you want to hold onto both goals, but increase the support given to the students on the writing prong?
If you choose the first option, you could offer something like a Technical Writing class, with a few group sessions and some one-on-one Teaching Assistant (TA) support, or you could ask the students to work with your campus Writing Center (if there is one).
If you choose to continue combining writing and oral presentations in one course, you could have them hand in the written write-up approximately halfway through the semester (but choose your date carefully, looking at the exam schedule), and do the presentations later in the semester.
Helping students become better writers is an iterative process, so plan accordingly and spread things out.  You could have multiple due dates, the first for the outline (this will produce better writing and will help students get in the habit of starting with an outline), one for the content, one for the bibliography, and then a due date for the final submission.
Regarding the mandatory meeting with the professor -- consider switching this to a meeting with a TA, and consider requiring two meetings rather than one.  (With just one meeting, both parties might be more likely to see it as a pro forma meeting.)  I would put the first obligatory meeting pretty early in the semester -- but after you've had at least one group session.
Examples of topics to cover in the group session:

How having an outline helps.
How to install a spell checker for a non-English language.
How to add custom terms to the spell checker dictionary.
Common examples of plagiarism and how to avoid it.
One or two examples of famous people who fell into the plagiarism trap through sloppy scholarship.
Walk them through an example of an "A" project and an example of an unsatisfactory project.
How many references should they aim for?  How do you decide what to include?
Later in the semester you can revisit some of these topics, once they've starting handing in some work; and also cover bibliography formatting and tools.

Overall, think in terms of supporting the development of better writers.  Sink or swim assignments that get a judgment (grade) might make sense for most homework and projects, but not for this.
It might be a good idea to spread the presentations out over several sessions, for example four sessions, every other Friday afternoon, with cookies.

Unfortunately the study regulations are very strict: The seminar must include a presentation on a scientific state-of-the-art result to a student audience as well as a seminar paper. We are just free to offer additional help or to implement an internal seminar schedule with several deadlines, feedback sessions etc.
You make a good point in suggesting that we go through examples of good and bad scientific writing and presenting. I think we can easily implement that.

@diplo - Not sure why you say "unfortunately."  Most of my answer is focused on the combo course design (presentations AND written submissions).

I was referring to the first paragraph of your answer. We have no choice but to stick to the presentation+paper course design because of the study regulations that hold for all seminars in our bachelor and master programs.

@diplo - That's okay, you can make the course work either way.  The main theme of my suggestions has to do with making the writing process more iterative and supported so that the students' writing skills improve.

I guess we are on the same page with that ;-). I just wanted to express that we cannot change the main goals of a seminar or drop the grading of the final papers. However we are free to implement extra tutorials on scientific writing, feedback sessions etc. (see your helpful comments) within the existing scheme.

Some of the following thoughts are already in the other answers, but i'd like to give a coherent answer, so please excuse any repetition. I've worked as a TA for several seminars which mostly were successful (and attended both good and less successful ones as a student), so here are some thoughts:

At my university (in Germany), there are normally two kinds of seminars. There are the normal seminars for higher level students, however before they get there, usually in the second or third semester, all students are required to attend a "proseminar" first. Outwardly, this looks similar to a seminar, however instead of focusing on current research topics, the talks are usually about something which is on the level of first year lectures but is normally not covered due to time constraints. (As an example back from when I studied computer science, I gave a pro-seminar talk on Petri-nets, which were not covered in the lecture on automata theory. Other talks in the same proseminar, were about different kind of state machines.)
Inwardly, the idea is on one hand to allow a greater focus on the presentation and the written paper, as the topics are easier and there are normally good textbook sources, but on the other hand without completely boring the students, as they still learn something new. (Compare this to a course on "how to give a presentation", which most students will hate or willingly ignore, since many, like me back then, falsely think they know better anyway. This also is a good reason not to tell them that the topics are not the actual focus.)
However something like this might be impossible to implement without reworking the curriculum.
Ideally you should know your students. This works best, if the seminar is directly subsequent to a lecture or if you have multiple seminars and "repeat customers". This way I often know, which students need strict deadlines and close supervision and which students can be allowed to roam free. Related to this, while there should be some choice in the topics, normally not all topics are created equal. Some are simply more complicated or have badly written sources, while others are mostly straightforward. Also not all students will have had the same lectures or even the same minor subject, so their previous knowledge of some of the topics may differ. A good way to handle things, is to present the topics to them and then allow them to give you a list of preferences (for example in the form of a doodle). This way they mostly get topics they are interested in, however you are still able to prevent people from taking on more than they can handle and you do not have to tell them that directly, as officially this is to prevent collisions.
Concerning the feedback, while peer review has its uses, it requires the peers to know what is expected. This works well for the actual talks, as the other students will notice if the explanations are incomprehensible or the slides are convoluted. However it works less well for the written papers or in order to gauge the correct understanding of the topic. This should be done by you and your colleagues instead. Ideally each student should have a direct supervisor assigned to them, who will be the main source of feedback and while in the end the professor has the final say, he will probably also be the one proposing the final grade on the written paper. This allows for more detailed feedback than if everybody has to read everything. You should however talk with the other supervisors regularly (for example at lunch) about what your students did and how you handled it, in order to have some consistency.
I always found it helpful to meet with each student one on one a few times, not only when they have questions, but also at certain milestones, where things are expected from them. An example plan would look something like:
6-8 weeks before the talk, after they have read their sources: Go through the sources with them. Let them explain the details, check if they understood everything correctly and help them with their questions, ideally also pointing them to additional sources. Ask them, on which parts they want to focus their talk, tell them what should definitely be in there and which parts they should skip because of time constraints and instead cover in the written version.
1-2 weeks before the talk, after they have prepared a first version of the talk: Go through their slides with them, try to correct problems and to estimate if the length of their talk is right. Ideally you should tell them to time a test-run themselves beforehand.
After you looked at the first draft of the paper: Go over the corrections with them. For the first draft I will usually purposefully waste a lot of red ink and mark everything I see, from factual errors and missing citations down to typos, bad formatting and formulations which I simply do not like and personally would change. At this point I normally will tell them that they should not worry to much, since this draft will not count for their grade, but that I expect them to correct all those and similar errors for the final version. Remember to keep a copy of the corrections.
After the final version is handed it: Give some final feedback. This is usually just a short meeting as most of the things will have been said beforehand anyway.
This is very labour-extensive, which is another reason to split the workload among multiple people. I usually spend around an hour per meeting and at least an hour on the corrections of the first draft. On the other hand, reading the final version is normally just a cursory check if they heeded the advice. However spending so much time with a few students also allows you to get to know them a bit better, which tells you if they will need additional meetings or even to give a test talk before the actual one. The general idea is always to correct problems before they occur.
If the topics allow this, it can be helpful to group the students into pairs covering adjacent topics or two halves of the same. Let them attend the meetings together and possibly even write a joint paper. This way they know enough about what the other is doing to catch problems and they have a vested interest to do so. It is not helpful to force students to work together against their will, however often enough, some of them will already know each other. A good way is to explicitly announce some of the topics as group topics and ask the students to state if they want to work together when asking them for preferences.

I am currently undertaking a masters year project in mathematics which sounds similar in structure to this. In semester one I had a talk to give at the end of the semester. At the end of semester two I will have to hand in my final "report" (approximately 40 pages) and give a presentation on my project.
For comparison, I have been having regular meetings with my project supervisor, where the main topic of discussion has been the report which I am writing. Here are my thoughts on your main issues:

With regard to the grammar issues, I think there are two separate things. Basic spelling errors can be fixed easily, as you say with a spell checker. Bad grammar is fixed by proof reading, but this is something which I can guess that the students are inexperienced with. Proofreading my own work is a skill which I have developed vastly during the time writing my project. I did not realize how much time it takes, nor how easy it is for mistakes to slip through the net. I suspect the students put a small amount of effort into proof reading but were not aware of how much time and care is needed to do this effectively. How you help them fix this problem is up to you. The problem was fixed for me by my supervisor relentlessly highlighting the mistakes which slipped through the net.
This must come down to quality of writing. What would be helpful for me in this situation (if I were one of the students) would be to have a "guide" piece of work to compare with. Particularly in the case of lack of explanation of basic ideas, seeing it done properly elsewhere would probably be effective. I can imagine that part of the problem is that the students are worried about "dumbing down" their work, and just thought that there were various things that didn't need explaining. It might be worth reminding them that they shouldn't assume any knowledge on the part of their audience.
Again, this sounds like something for which it would be useful to have a comparison. Perhaps in this case you could provide the students with a model "bad" example, and a model "good" example. That way, they can see which is closer to what they're doing. Giving examples (like the one which you have given us) of imprecise language and why it's not appropriate would probably be helpful.
This sounds like inexperience on the part of the students. It is also (to my mind) indicative of a lack of wider research on the part of the students. Is this something which you have told them is important? Students are lazy.

I think generally, an iterative feedback process is extremely helpful, and would fix many of the above problems. That is because that has been effective in my case, and a lot of the mistakes which you are pointing out resonate well with me (particularly points 1, 2 and 3).

In my experience the problems you face are very common and I believe that it is almost impossible for the students to learn all the necessary skills within just one course. In order to address these problems, together with colleagues of mine, we tried to introduce parts of these skills in a variety of courses starting with the introductory courses.
In our study programme there are multiple exercise classes where the students are supposed to solve problems and to present their solutions. Traditionally their presentations were graded only based on the quality of the solutions themselves. For these courses we introduced a grading scheme for these presentations which take the quality of the presentation into account. This grading scheme contained different parts which address different aspects of the presentation, e.g. whether the problem was explained or whether the necessary background was presented.
Since these students also had to hand in some of their solutions we introduced a similar grading scheme for these solutions which again took into account the quality of the exposition.
In more advanced courses we combined these grading scheme with some kind of peer review, i.e. the students had to grade some of the "papers" and some of the presentations.
In our experience, these exercises improved both the quality of the students' writing and of their presentations. It also seems that the existence of an "official" list of criteria for a good presentation or a good written text already improved the quality.


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How can I compare photos and PDFs submitted for an online math test to detect for plagiarism?
I have a group of 1000 students taking an online math test, and 10 teachers (each one evaluating 100 students). Some of the students sent photos and others sent PDF files for their submissions. I already found 2 students that sent the same document (yes, exactly the same), so now I want to compare all documents.
Does there exist any way to compare all the documents to find similar documents?

Such software exists but I don't think it is free. Ask the institution you work for if they have a subscription. In any case, it would help to require the students to submit a file-format that is easy to process, such as pdf-files and not photos.

Finding identical files in a directory is simple. Five minutes with python or a similar script language.

What is the content of those "evaluations"? Multiple choice?  Then you cannot prove anything from identical answers. Written text? Do you have a system to OCR a thousand documents?

Are these student evaluations of the course or are they something else, such as examinations? It is unclear.

The evaluation is a "typical" mathematical test, but on-line. The first part consist of alternatives. And in the last question the students must to submit its calculus. The ideia is to compare documents (pdf and images) and detect evident plagiarism only.
@Karl do you have a concrete ideia? I have a good pc and also a linux server in the math departament to run this kind of stuffs. I'm good programming too. Btw, the only word "OCR" is very usefull to me, now I can search on google how I can compare documents.

Looking for identical images will catch one or two, but try running a google search with a fairly unique phrase from any one of your questions, all inside of quotes, and then the word “Chegg”. What you find may encourage you to check for more than just identical PDFs.

First thing you should have specified is that all students submit using pdf. images not accepted.

I'm probably a bit late to the party here but still going to post an answer for posterity.
My specialty is not in image processing or OCR (Optical Character Recognition) but I'm still able to throw some advice out here. Furthermore, I am going out on a limb to assume you are looking for free (or DIY) solutions given that there are readily available products out there.
So my first course of action would be to find a way extract computer readable data from PDFs and photos. OCR would come in handy. However, in this case because it is a math exam it would be complicated by the mathematic symbols. Tesseract used to be my go to OCR engine (in 2015) because it is free and readily available. I am unsure if now they have added the capability to parse maths symbols but I'm sure there are other engines out there that have this kind of capabilities. Worst case scenario you can omit the symbols and only extract the numbers.
The next one would be to find a way to compare similarities between them. A naive way would be to employ string similarity algorithms (Levenshtein, Cosine etc), which would catch the really obvious ones. The next step up would be to define some abstract data structure (for example, a syntax tree) to encapsulate the structure of a particular solution, then find a way to compare whatever structure you came up with. For example, when I was working as a tutor (TA) for a database course, I converted SQL queries into a AST (abstract syntax tree) and compared the similarities in structure in order to filter out potential plagiarisers.
The final step is to actually run this program at scale. The following is based off of @yemino's description so may not apply to the reader's particular circumstance. Based on what you said, I'm assuming you have access to a compute cluster (i.e a collection of Linux servers). The easiest way I can think of is to setup one server in your cluster as a LB (load balancer) and to simply RR (round robin) route all tasks (where each task is an individual comparison between 2 submission) to the rest of your servers. If it is one giant instance, you can dockerize your program and spin up X number of docker instances (depending on the capability of your instance). Then take the same principle as above, spin up a LB and RR route the tasks to your docker instance.
Addendum: As of the date of writing this post, I explicitly avoided Generative AI mostly because a) they are paid and b) their CV (computer vision) capabilities are not rigorous enough for me personally to justify using them for something like this.
Addendum #2: Machine learning solutions are also possible but probably impractical due to the amount of training data required which is why I wouldn't recommend going down this route.

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Faculty job negotiation - dual career and salary
I received a phone call from the department chair informing me that they are willing to give me an offer. Then, he sent me the startup sheet and asked me to send them my comments. This sheet contains salary, student support, travel, moving expenses, and a few other items. The chair also asked me to add any equipment I need and he made it clear that he will push as much as he can because it will be beneficial to the department whatever I can get from the university for my startup in terms of equipment.
For this position, I plan to join the target university (R1 university) as an associate professor, and thus the salary is a bit confusing for me since the salary for other department members is not online. My current institution is giving me, let's say $100K but they also give me 20-30K in the summer to stabilize my salary due to declining my previous request on raising my base salary during COVID. However, the destination university is giving me $120K as my base payment [these numbers are just random numbers :)].
I have two questions:

I am not sure how I should talk about my salary. Because the base payment of the destination university is higher but the total payment is lower!

I also need to ask them about my spouse and her employment. I am not sure if I should bring this up now or should wait until I finalize my offer and then ask them about a dual career. I am not sure when the right time is. If I bring this up now, then, they might not fully consider my requests on salary and other items. If I also bring this up later, they might say why I have delayed it since spousal hiring needs a long process and discussion. When do people discuss this matter based on your experience?

You've made this awfully individualized, making it difficult to give advice. If your spouse's employment is a vital concern you need to consider that. If money is the deciding factor, ... etc. It is your own values and needs that you need to think about along with your spouse.

@Buffy: Are you saying that I should bring up both the salary and my wife's employment at the same time so that they can make a balance?

Actually, no. I'm saying you need to consider your needs. I can't do that for you. But it is probably a mistake to try to finesse it too closely. Having requirements that you don't reveal isn't going to ease the way. It isn't poker.

I agree with your comment. But the issue is that I don't have any written offer yet, it is just a request to finalize some numbers for the offer.

If the spouse's employment is a deal-breaker I would think you would need to bring that up sooner rather than later.

Is it typical to get a written offer and then negotiate on that, or do some of the negotiation before receiving a written offer?

As a general rule, if you go into details discussing issue X before you even mention issue Y, people will reasonably assume that X is more important to you than Y is. Now think of the special case when X stands for “salary” and your spouse’s employment is the Y. Is it really the case that you think it’s more urgent to discuss salary than something as critical as your spouse’s ability to work?
About talking about salary, I think you may be falling prey to the chair’s trick of asking you to send “comments” about an offer that doesn’t yet exist. This is a common trick that employers pull (more common outside academia, but I’ve seen it within academia as well) to get job candidates to disclose part of their internal bargaining parameters. It’s similar to asking a job candidate the question “how much did you make in your previous job?” (now outlawed in some places), or “what are your salary expectations?”. The professional way to answer this is “I am really excited about the position and am looking forward to receiving an offer from you”. In some situations it might be beneficial to give hints as to what might be an acceptable salary figure, but there is a risk in doing so, and the precise conditions when that could be a useful thing to do would be extremely situation-specific and probably not relevant to your case.
In any case, when comparing salaries at two different places, what usually matters to people is the bottom line, not which part is called “base salary” or any other name. Of course, the cost of living also varies from place to place, so your comparison of the offer to your current situation should take that into account. Good luck!

First off, determine your bargaining position. Do they really need/want you or can they just get the next one on the list when the negotiations don't work out? Be realistic: for most associate prof. jobs your negotiating position is very bad. So you can't get everything you want, and probably you can't get much. So you really need to prioritize and decide what is the absolute minimum you are willing to accept and when you just walk away from the negotiation (if ever, but then you're negotiating position is pretty much non-existent).
In my university, if the negotiation is over, then it is over; the university will not negotiate dual career options after the negotiations are over. So you need to find out if that is also the case in your university. If your university works like mine, then you have to bring the dual career option up at the beginning and accept lower pay (if you find the dual career option more important than your pay).

Thank you for the advice. Given what you have mentioned, you are saying that I should negotiate all these items together, not separately. The issue is that I don't have any written offer yet, it is just a request to finalize some numbers for the offer.

That is not what I am saying. I said that you need to figure out how the negotiations happen in your university.

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Academia or industry: which is easier to begin a CS research career without a bachelors degree?
I'm an engineer from the US at a large tech company who had to cease their undergraduate studies due to family complications. Previously, I was an assistant in a CS research lab during college, and in industry I've been able to work on engineering problems within the same CS sub-field for the past 4 years. I was a co-author on a published paper while working with the research group, though the professor I worked under has since retired.
Increasingly frequently, I have been reading papers, emailing authors, and desiring research related work again.
If I want to work towards a research career again in this sub-field, and don't want to complete a bachelors degree, what would be the easiest or most realistic path forward - entering a research role at big company without a graduate degree, or beginning a graduate degree without a bachelors? What are possible long term drawbacks of each path?

Where are you? Country?

Entering a graduate degree without an undergrad is possible though not easy. The best path there is to accomplish something on your own and attract the attention of somebody with influence. It would need to be substantial.

If you are concerned about long-term advancement, income, respect, being "one of us", etc., then get the degree.  It's obviously not like this everywhere, but in many places there will be an (often subconscious) attitude that you are different, and expectations will be different.  Compare it with being a female employee in an all-male department.

@Buffy US, thank you for the ask, I will add it to the post.

I appreciate the insight @puppetsock, my professor may be able to recommend me when applying to a program, but I am doubtful his recommendation would cover an edge case like mine. There are other SE questions about doing entering a graduate program with no bachelors, but none seemed to lay out actionable advice other than how exceptional it would have to be.

As you mentioned @Ray Butterworth, I am worried about losing 'agency' in my work if I went without a graduate degree. Even though I had success in engineering without a bachelors, I assumed the environments or attitudes would be different for researchers, so thanks for your insight on that.

I don't think there is a good yes/no answer to this unfortunately, and by that I mean different companies and university have vastly different rules/traditions/standards as to how they hire/admit people to programs.
I would say the easier of the two would be entering a research role at a company with just a bachelors degree. The reason I say that is because what is more important (in my experience) at corporations is relevant work history/experience.
I would note that in my experience companies can be very wary about hiring people without at least a bachelors degree because they can be seen as a liability.
I have never heard of a CS department admitting someone to a research role without a bachelors degree. I would guess that this is because it is rare having someone want to research without a bachelors and on top of that it is already rare to award graduate degrees to people who don't already have a bachelors. Like I said in the beginning though, you may find a program willing to accept you.

Having a BS/BA is not your issue, actually. Even in industry, great research organizations are staffed with PhDs and equivalent. What you want, if you feel the need to leave your present employ, is a doctorate. That will make you desirable. But you may have worked your way into a comfortable niche and it may continue to work for you without further action.
But if you want to gain the ability to move about freely in the world of research, either in academia or elsewhere, then you need to find a way to get a doctorate. I would suggest that you try to do this without going back and starting with a Bachelors, especially in the US. You probably already know whatever it is that a bachelors will teach you. This is especially true of knowledge required to do a job, but may be missing some theoretical parts of CS. The missing pieces may be more or less important depending on which sub-field you wish to focus on.
My suggestion is that you go and visit a few places, after making some appointments. You want to talk to a few faculty members. You want to explore your options with them. They can tell you what you need to know to start out with a combined MS/PhD degree program. You can tell them what you already have skills with. Perhaps they match up well.
Such a program in the US normally starts out with quite a lot of advanced coursework. It assumes the knowledge of a bachelors, but not much else. If you can handle that coursework then there is no issue about your suitability. The coursework leads to comprehensive (qualifying) exams. If you can pass those, you are considered ready to do research.
Any program that would accept you would be one that is fairly large, but not so large as to be required to fall back on rigid rules of entry.
You might also be able to convince some place to accept you through a small number of exams. Perhaps oral exams in which someone explores your knowledge. This could also give you a way to learn what actual holes in your education are required to be filled to be accepted into a full doctoral program.
But I think that completing a BS/BA is not necessary and would be the longest path to getting the needed credentials as a researcher. You would find much of it boring.

In industry, with your background it sounds like you are just as well-positioned to succeed as someone with a bachelor’s degree. But if your hope is to get a “research role at big company”, that sounds very unlikely whether or not you have a degree, because industry has very few (almost none in fact) true “research roles”. They might have roles that involve one or another thing they call “research”, but since you mention wanting to work towards a “research career”, I don’t think what they usually mean by “research” is what you’re thinking of. (True story: a recruiter once called me up trying to get me interested in a “research” job on Wall Street, and had a hard time understanding why I wouldn’t be interested in switching from a “research” position in academia to a “research” job in the private sector that pays much more - after all, research is research isn’t it...?)
As for academia, the answer is also probably not what you were hoping to hear. Without a graduate degree, you have zero chance of finding an interesting job in academia that has a significant research component. (In fact you’ll have a hard time getting any job without a bachelor’s degree - academia is kind of conservative/snobbish about degrees.) And without a bachelor’s degree, you’ll have only a very small chance of getting into a graduate program - certainly not large enough to make this a reliable plan for career fulfillment, although if you wanted to try anyway, Buffy’s answer has some interesting suggestions for how to go about it.
TL;DR: if you want a research career, finish your bachelor’s degree and then go do a PhD like everyone else who wants a research career does. It sounds like you have the talent and abilities, so hopefully you can find a way to make it happen. Good luck!

Is the comment on industry research true for all of CS? Aren't there big companies that do serious research and publish in top venues in fields like machine learning for instance (Google, Facebook, Microsoft, etc.)?

@GoodDeeds I didn’t say there aren’t any research jobs in industry. There are just very few of them. And I suspect most of the researchers who “publish in top venues” at those companies actually have a full time job doing practical engineering work, and publish papers in their spare time or as a very small portion of their employment time. But I am open to being corrected by someone who’s more knowledgeable about such things.

I believe they have dedicated wings that focus on and hire people solely for research (e.g. Google AI, Facebook AI, Microsoft Research). Some of the researchers hold dual appointments as professors at universities. They have also made major breakthroughs, like Resnets in computer vision.

@GoodDeeds there is some truth to that. That falls in the “very few research jobs” category. And as you come close to saying in your last comment, many of those jobs are roughly equivalent to academic faculty positions, and require the same sort of qualifications to be competitive for. So I wouldn’t advise OP to bank on getting such a position without a graduate degree. It won’t happen.

Yes, I agree with that.


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Are scientific contributions of the article listed in the abstract?
As the title says. To put the contributions (explicitly) on the abstract:

Is considered good practice
Irrelevant as long as you correctly describe your paper? Source Research Guides
No, explicitly mentioning the contributions is not considered good practice and should be avoided.
Other?

Here is an example of a paper I published on ArXiv. Would you say that the abstract is Ok? Please see I put the main emphasis on the contribution (I use the contribution as an excuse to then explain all the work done).
NOTE: Please don't confuse with the contribution of each author to the article! An article should have something new, therefore we can say a contribution to science.
Some article's have a specific section for that reference. Others just mention it reference but my question goes if it is good to mention it in the abstract or not.

As this question is somewhat subjective and there might not be a CORRECT response. If after some time I consider fit there is not a precise justification (some indisputable reference for example) I will mark as a solution the response that had more upvotes, supposing the upvotes reflect what the community thinks.

Just an opinion, but I think there are more important things that need to go in the abstract.

@Agustin I guess I am a bit confused. How could you possibly write an abstract without noting what the paper brings to science?

That is indeed a very good point. I believe the correct thing to say is explicitly saying what the contributions are. In the abstract also, there might be things that there are no contributions. I edited the question to be more clear.

In light of the example you linked, you should indeed state your contributions, but there is no need to state that these are your contributions. So instead of stating "The contribution in this paper is considering problem A using method B", you should use "We consider problem A using method B". Doing this would have easily saved 20 words in your example without changing any of the meaning. Even more importantly, the useless words are in the beginning, which is the only part of the abstract you can actually be sure that people will read.
The default assumption on a paper is that you are presenting something new and that new thing is why people should be reading your paper, which is why will be prominently mentioned in the abstract. So the default assumption when reading an abstract is that all results mentioned describe scientific contribution. There is thus no need to explicitly mention it.
There are of course things in the abstract that are not new contributions, but they are generally made obvious by phrasing. If your abstract starts with "We survey some recent results in ..." no one expects that the results mentioned afterwards are new. Or if you are (in)directly referencing using "The well known problem of ..." then no one will think you came up with the problem. But if a sentence starts with "We show that ...", then any reader will assume that you claim to be the first one to do so in this context.

The abstract is there to let you know what the article is about and entice you to read it. I would want to know what its claimed original contributions are from the start.

Personally, I like abstracts that roughly follow a 'Background and Aims, Methods, Results, Conclusions structure'. Contributions are meant to be implicitly written and I am not a fan of explicitly writing these within the abstract. This is still a personal decision and is therefore completely left to the author.


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Dual job within German university with 25% contract
If someone currently working 100% with employer 1 in a public job. Is it possible to get employed by a university or public organisation (employer 2) with 25% or 35% contract after having a Master's or Doctorate degree?
The dual job includes engagement in some activity within the group.

Please clarify what exactly you mean by "dual job". Do you want to be employed by two different employers, working 25 % or 35 % for each? Generally, all kinds of part time contracts are allowed.

I have edited the text, please relook it.

What does the sentence "The dual job includes engagement in some activity within the group" mean?

Look into regulations for "Nebentätigkeit" applying to both employer 1 and 2. It's generally not uncommon but you have to follow the rules regarding time extent, approval, etc.

@silvado thanks for your answer, in practice, have you seen it? I couldn't explore it much.

I'm not sure about the specific constellation that you are interested in, but the first point to go to would be the personnel department at employer 1. If this is a public institution (university, research institute, or the like) they should have some guidelines about side jobs (Nebentätigkeit) that you need to follow. The specific regulations may depend on the type of your appointment (TVÖD, Beamter, ...). If this is a larger institution you can probably find the applicable guidelines on their intranet (possibly in German only).
If everything works out with employer 1, you also need to check with employer 2 which regulations apply.
For professors at German universities, side jobs are generally possible (subject to specific rules) and are being used. I know cases where the side job was in industry or other universities abroad. I don't personally know of a case where the second employer was another German public institution, but I think it should also be possible.

@silvaod, thanks for the detailed description; I agree with you on the overall description.  Only the last paragraph raises the question of whether a professor is doing a side job in a private company or running their own company. The number of hours is more than 9-10 hours, and even it's unofficially. This is my exact question.

@GautamShahi The professor will have to ask for permission before starting the side activity, and normally if it's beyond a certain time limit (probably 20% of the regular working time) it would not be allowed. I think this will also apply to running an own company, but you'll have to check the regulations that are relevant for you, and, if in doubt, ask you administration.

@silvado: the university (or the Land) may decide that the 2nd position is in their interest and free their employee/Beamer for x h/wk. While I don't know for sure, I think this is how people who are professor or teach at a university plus have a position at a non-university research institute do it (I know several such people). (Works also on lower level: when I was employee of the university during my PhD and got a scholarship, the university decided that it was in their interest to allow me to have the scholarship, and we agreed to have the employment contract "hibernate" for that time)


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How to give opinion to my professor about writing LoR?
I have been applying to PhD positions (got offers already, thanks god) and surely I asked my master thesis supervisor as one of my letter writers.
He is quite new as a PI. He told me that it is his first time to write a recommendation letter for PhD applications.
I had a really good time working with him and believe that he is nice and thinks good of me. (of course, he may secretly think that I am stupid).
The problem is that I actually received a feedback from a position that rejected me suggesting that my reference is not very strong. On top of that, (maybe just by coincidence), I didn’t include his letter in my successful applications.
As he is lack of experience, I doubt that his way of writing the letter may be too subtle or not giving enough details or such. (Again, of course, he may secretly think that I am stupid.)
As it will possibly be good for him and any of his later students, should I hint him about this? If yes, how should I say it nicely?
Thank you.

I have two suggestions. The first is to give him an outline of what you think your accomplishments are and say something about your enthusiasm and future goals. Or, at least, offer to give him such an outline if it would help. This is just to give them a few ideas.
The second, probably more important, is that you ask another, more experienced, professor that is also writing for you, to speak with the advisor and also give them ideas about how to frame such a letter. It is normal for one professor to help out another, newer colleague, on their responsibilities.
I'll note that some letter writers actually ask the candidate for a first draft of the letter they will send. There is some controversy about this, but it is common enough that you shouldn't be surprised if it happens.


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When is the earliest a US university will create an official transcript?
This question about not supplying a transcript from a university they just started attending in an application to a different graduate school got me thinking. At what stage of being a student will a US university generate an official transcript.
When is the earliest I can get an official transcript:
After being admitted, but prior to registering for classes?
After registering for classes, but prior to starting classes?
After the start of classes, but prior to full refund drop date?
Some other time?
Presumably this information is buried in some webpage of the university registrars office.

Are you asking about official transcripts specifically? My recollection of campus software I used as an undergraduate is that unofficial transcripts could be easily produced at any stage as long as you had an active university account, which would be somewhere between admission and registration.

@BryanKrause yeah the official fancy one (see edit).

Is there any particular reason you could not ask your registrar this question?  (Horse's mouth, and all that....)

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Is it okay to get partial funding for a conference from two institutions?
Is it possible (and legal) to get funding from both the host university and the university the student is from, for overseas conferences?
To elaborate, let's say I (a student) am going for an overseas conference. My home university is able to fund me (but only partially). The host university is also providing funding (but limited funds, so I am assuming it is also partial funding). Is it possible to apply for both these fundings?

I would say you could apply, yes, but you should mention that you also applied for the other funding. To make sure that you don't run into problems, I would first discuss it with the responsible people at both institutes, as the rules might differ from conference to conference, institute to institute, depending on country, etc.
Then, I would write down a short statement of cost. For example:

Flight:  500
Hotel stay: 350
Conference fee: 100
Meals (if not included in hotel cost): 50

Total: 1000.
For this cost, student ... received partial funding from institute A: XXX and from institute B: YYY.
Make sure that $XXX + YYY \leq 1000$ and then have both institutes sign it. This should put you on the safe side, as you can use it to prove that both institutes knew about the other partial funding and that you weren't "overfunded". Of course it would be best to have an estimate of the costs beforehand and to discuss these things also before you receive money, so once again: Go talk to the people responsible.
And I hope that it's clear that you should, under no circumstances, get overfunded here. Don't risk it for a few bucks, this might strongly backfire once someone finds out.

This is certainly a safe approach, but maybe unnecessarily cautious.  If one institute offers you partial funding, I think they would assume the rest is probably coming from some other institution.  Typically you sign a form saying you are not being doubly reimbursed and that is the end of it.  Of course, you should save the records of your expenditures and reimbursements just in case anyone asks in the future.

@NateEldredge - That would make a constructive answer.

I have done this before and did it basically exactly as described.  I actually had three funding sources (one fund at one university, and two separate pots of money at another) and it went fine.  It's easier, I feel, if they're collectively offering far more than you need, so you can get the flight from source A, and the hotel and conference fees from Source B.  Less paperwork than if you need to split an expense between the two sources, but may not always be possible.

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What could/do applicants usually ask professors, and how does this reflect in a statement of purpose?
When applying for PhD programs (or MPhil or MS) that require only a statement of purpose and not a research proposal, I was advised to first check if some faculty members (or is one good enough?) in the university have the same research interests as I do. I remember reading or hearing something like, it may be helpful to read up on the research that those faculty members have done and possibly contact some of those faculty members, and then incorporate the reading and contacting in the statement of purpose.

So what could I ask those faculty members? What I think:

Hello Prof/Dr ___ .I'm interested in applying in your univ and am interested in X. Can I ask more about what you research? This is my background (I'm from blah blah blah. I have a master's, but it's in applied not pure math. I've done some self-study blah blah blah). Do you think I am a good fit for the university? Would you consider taking me on as a PhD student? Do you think I am better suited for MS/MPhil than a PhD?

And then how do I incorporate the e-mails in my statement of purpose?

I'm interested as well.

@BCLC Thanks for setting the bounty! Good luck with your applications! :)

If I were a professor, I wouldn't accept a student who doubt his own suitability. I'm not gonna do any psychological counselling.

@Michael Lolwut?

It's fair to expect that a student who doubt his own suitability would have some psychological problem during his pursuing a doctor's degree. Then his mentor would probably have to do some psychological counselling stuff. To avoid this, the best policy is not accepting the student at the beginning.

Can I ask more about what you research?

Don’t be so vague. You can find out for yourself what a professor is researching by reading some of their most recent publications. Think about the papers carefully and try and ask a pertinent question, for example:

I was reading your paper X and have a question about Theorem Y. Is it related to Lemma Z as described in paper Q?

Do you think I am a good fit for the university?

Instead of asking what the professor thinks, I would emphasise why I know I’m a good fit for that university. If you don’t like the look of the department, why bother contacting them in the first place?

Would you consider taking me on as a PhD student?

I prefer “Will you be taking on a PhD student next year?”, although this may be just a cultural/ politeness thing.

Do you think I am better suited for MS/MPhil than a PhD?

Again, I feel this is something you need to decide before contacting them. Look closely at the program requirements and think critically about your own skills and knowledge and where you will fit best. If you’re still in doubt, contact the admissions tutor with this question, not a potential supervisor.

Many of the PhDs I’ve been applying for lately strongly encourage inclusion of a potential supervisor's name in the statement of purpose, so they can direct the application to the relevant person. You could include a mention of the professor you contacted in the following way:

My research interests lie in the field of pure maths, and more specifically how Lemma Z can be applied to Theorem Y. I would like to work with Professor A in research group B.

All of this is based on my own experience of applying to UK physics PhDs (which I am still in the process of doing).

Hey Natalie Hogg, great answer. I have some questions about your answer. Perhaps, you, OP and I may benefit: 1a. Okay so how does your suggestion fit into the SoP? 1b. same question. 1d. How does an admissions tutor know better about for example string theory than a string theory professor? 2. Instead of 'pure maths', why not a specific branch like 'numerical solutions to PDEs' or 'stochastic analysis' ? Are you going to say your research interests lie in physics/applied physics rather than a specific branch like 'string theory' ?

1a and 1b: I would not include any other mention of a specific professor except in the way I described in part 2 of my answer. They should know why you're applying to their department based on the research interests that you have. 1d: In my experience (possibly UK only) the admissions tutor will be a lecturer or professor in that department. They just handle admissions too, so they know as much about string theory as any of their string theory professor colleagues.......

...... 2. This was just an example, I don't know all the ins and outs of pure maths. In my applications I am writing something along the lines of "my research interests are in theoretical cosmology with a particular focus on inflation and dark energy". However, I can't write your SOP for you. You must decide how best to word it, and this may vary between fields.

From my reading here at Academia SE, this kind of self-recruitment letter may be appropriate in parts (or much) of Europe, but in the U.S. it is more common to simply apply for admission to the department and then match oneself up with a specific advisor after at least a year of student.  (There are exceptions, of course.)

@aparente001 Not in Country A. Thanks and thanks astronat!

astronat, @aparente001  what do you think of Barry Rountree's 80% hit rate email please? I'm posting a separate question about it.

I think you have misunderstood the advice you were given.
When you are advised to make sure your research interests align with those of some of the people working at the place you are applying to, people typically expect you to do the work yourself, not delegate it to random faculty members.
If I received an email asking me what my research interests are, I'd either bin your email or, if I'm feeling generous, redirect you to my easy to find personal webpage where I've listed my research interests in great detail.
What you should do is go to the webpage of the institute/school/centre/unit/whatever you are applying to, and look through their staff directory. For each faculty member, there will often be a short summary of their research interests, and/or a link to their personal page, and/or recent/important publications (publications can be found in many other places as well, e.g. Google Scholar). If you cannot figure out the research interests of a faculty member using any of this information, perhaps graduate studies are not for you.
Or if that's too tedious for you, you can always look at research groups first to narrow down the list of faculty members. In this case, go to the webpage of each research group affiliated with the place you are applying to, and there should be a summary of the group's interests/goals somewhere there. If that information is not enough for your needs, but so far you think that group may be a match, go through the staff directory of the group and check the research interests of each group member. And if you want to know the specific things the group is working on, check the group publications.
So, contacting faculty for research proposals is inevitable, but completely unnecessary for research statements.

@JackBauer - Although this answer is a bit blunt/tactless, the procedure outlined is good.  From what I've seen of your participation at Academia SE, you should not have any trouble following this outlined procedure.

"Can I ask more about what you research?" is different from "what my research interests are" ? Thanks 101010111100 and @aparente001!


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Is it common for an undergraduate thesis in pure mathematics to prove something new?
What do undergraduate students in mathematics do for their thesis, if they have done one, besides expository or applied math?
I was thinking that the kind of research they do is something applied, say using math in social sciences or a problem in one of the less rigorous natural sciences, or discussing such a problem (that's what expository is, right?).
To me it seems something non-expository or non-applied is an original contribution to mathematics, something that PhD students do.
I attended some pure math undergraduate thesis presentations. I was quite surprised: Did they prove anything new? Never bothered to ask due to fear of looking stupid. Would it be out of the ordinary to expect an undergraduate proves something new? If they did not prove anything new, what the heck are they talking about?
It seems like if it's not new, they are giving a lecture. If it's new, that seems like a PhD-level accomplishment.
I mean, do math undergraduates frequently prove new things?

"Anything new" is rather broad. I myself proved "something new" in by bachelors thesis, in the sense that nobody answered that particular question rigorously before. Was it deep? Probably not. Could I have published it? I don't think so. Still, it was new.

Depending on the country and the quality of the teaching, yes it is possible. If you have a Professor who gives you an actual problem knowing you have been taught the right modules/topics to investigate it, yes. If you have little teaching and then are told to pick a topic (as it happens in some places) then the chance is significantly lower.

@DetlevCM I'm guessing there may be 1 for every dozens or hundreds. I was actually wondering about the average batch of undergrad math majors whose thesis is in pure math. My guess is in the first place not many math majors will do pure math in their thesis. So what about those who do? They actually try to prove something? What happens if they cannot prove that particular conjecture in a month after they come up with the proposal after a month? 2 months left in the semester. So what happens?

@paulgarrett Thanks. Technically my question is about proving new things not doing "research" (in the professional sense). I was just thinking that doing such things is very difficult without a master's or PhD. I guess it's not impossible so what I'm wondering is: What the heck are these math majors discussing in their pure math theses? Anyway, time to read the answers

@JackBauer This come back to good supervision - a good supervisor should notice a given problem is too difficult and should also point a student in the right direction if he notices the student is too ambitious.
On the other hand, if you can show a comprehensive piece of work detailing or investigating a problem, this may prove enough for especially an udergraduate degree - even without a novel contribution.
(Just on a personal note, I know of someone who wrote an Enigma simulation for their BSc, something I did throughout the year as an assignment in a module...)

@DetlevCM Ah, that explains it. Thanks. That sounds cool. Enigma like in WWII?

@JackBauer Yes, that Enigma - I only covered the rotors and left out the switchboard but the switchboard is the trivial part. Heck, I suspect I could write an implementation fairly easily nowadays having gotten better at programming since. (Side note, its not cracking Enigma, its encoding and decoding which is really trivial.)

@JackBauer Something like that. Some researchers claimed a property of the things they worked with; it was crucial for their method to work, but they did not provide a proof. (I don't know if they could have.) I filled that gap. My advisor found it, and I had the luck that it was a reasonably scoped task that took mostly undergrad stuff plus some tinkering. (I think he (and I) hoped I'd happen upon a case where they were wrong, but they weren't.)

I'm going to disagree with Oswald. In my experience, undergraduate students do not often prove new things in pure math. I wouldn't even say master's theses often contain new results. There are a few main reasons for this.
Firstly, pure mathematics operates at a level that is not very accessible for most undergraduates, even those doing research. Undergraduates doing research are often well out of their depth and holding on for dear life. This can mostly be attributed to just not having enough time to get up to speed with what is considered modern mathematics. Most courses in mathematics at the undergraduate level are about math from 50-100 years ago (if not older).
Secondly, undergraduates do not often have the mathematical experience to know what the right plan of attack is when faced with an abstract and new problem and they may not know how to check their work thoroughly to make sure there are no major oversights or blunders. A lot of mathematics involves lateral thinking and it takes a lot of time to build those connections. The hardest part of a pure math PhD (in my opinion) is learning how to attack a problem no one has considered before. Standard techniques that others used may not be useful at all to you for one reason or another. An undergraduate won't have the creativity to navigate this kind of issue because the kind of creativity that is needed comes with a lot of experience. Even when an undergraduate student thinks they've proved something, the nuances of their argument likely will not be apparent to them. (This is especially true when it comes to functional analytic/measure theoretic arguments - the devil is in the details.) Thus a proposed proof may not even be close to being right.
Lastly, not many undergraduates in pure math do research because the gap they have to overcome between coursework and modern mathematics is pretty substantial. Those that make contributions in pure math are those that are very, very talented and have very thorough backgrounds (backgrounds that rival master's/PhD students).
Undergraduates in pure math are not expected to make contributions. That is not what research is about for them. Introducing an undergraduate to research serves a couple of different purposes: it introduces them to more advanced topics and it gives them a taste of what research is like so that they can make an informed decision about whether or not graduate school is right for them. As such, the theses are more like surveys of a specialized topic in mathematics. There is a lot of independent learning involved and there may be some unique examples, insights, and connections contained therein. They may not be presenting "original" work, but poster sessions are there to present what they've learned regardless of whether or not it was original. So yes, it is kind of like a lecture. They are undergraduates and far from being experts in their field.
Note that I am not saying that no undergraduate ever produces new results in pure math (there are some high school students that are better than most PhDs), but it is not a common occurrence and is not expected or considered the norm.

Bingo. Exactly. Further, I think it is bad to promote the mythology that "undergrads can do meaningful research in mathematics" if only because it sets of unrealistic expectations, so that "everyone fails". That is, it does not help anyone to "assure" them that "they can do research while undergraduates", because most likely they will not, and this is not failure. And so on. For that matter, many graduate students misunderstand the degree of "originality/creativity" that will actually play a role in their thesis, since the bulk of the work is assimilation of known techniques...

I think a large part of the difference here is subfield.  It is very rare for an undergraduate to make a substantial contribution anywhere, or any contribution to a subfield requiring a large amount of background.  On the other hand, it's not so unusual for undergraduates to be able to prove new results in many areas of combinatorics, even if these results are unlikely to be interesting to anyone except other undergraduates working on follow-up projects.

I fully agree with @AlexanderWoo (and, perhaps counter-intuitively, Cameron's Answer): I think undergrads can definitely do bona fide research, in combinatorics if nowhere else. But, it is probably is likely that most undergrads don't do original research.

@AlexanderWoo I was definitely thinking mostly of analysis, algebra, geometry, and set theory when considering pure math, but combinatorics definitely lends itself to undergraduate contribution. The others are almost impossible to get into as an undergrad unless you're effectively at a PhD level already but that is a very small portion of students.

@Alexander Woo - I think it is important to distinguish between undergraduates working alone (who are indeed unlikely to produce much publishable work) versus undergraduates working in collaborations with faculty. For example, the well-known Duluth REU run by Gallian states they have over 200 published papers, in professional journals. These papers seem to be no more likely to be "uninteresting to anyone" than all the other papers in those journals :) See http://www.d.umn.edu/~jgallian/progbib.html

Let's face it - most math papers, my own included, are pretty much uninteresting to anyone.  They're written at least as much for the ancillary side benefits of research activity to education as for any importance in what they contain.  I stand by my statement, even for most (not all!) papers produced at Duluth.  (BTW, there is a lot of undergraduate research available in metric geometry - I suppose you might consider that combinatorics rather than geometry though.)

I do agree that most math papers are not of world-changing importance. But I think it is too much to hold undergraduate research to a hypothetical high standard that most papers by professional researchers don't meet. If an undergrad is author or co-author on a paper in a journal that a colleague also publishes in, I think we have to count that as genuine research by an undergrad. Random unpublished "senior project" type work is something else entirely, of course.

Thanks Cameron. Please expand on "As such, the theses are more like surveys of a specialized topic in mathematics." What do you mean by surveys? The heck they are discussing is precisely what I am wondering.

@AlexanderWoo Combinatorics? Elaborate further please. I cannot imagine any area of pure mathematics that is regularly contributed to by undergraduate research.

@JackBauer By surveys I mean they're expository rehashes of some topic in math. Often it isn't easy math by any means but it may be a topic that hasn't very many elementary introductions. The survey might act as a basic introduction for people looking to get into the topic. Often novices to a topic provide a very different and unique insight and can deconstruct ideas in simple ways (because they have to in order to understand it), so it provides a gentle exposition.

@JackBauer - see the link that OswaldVeblen provided.  All those papers were written by undergrads.  Personally I coauthored an REU paper as an undergraduate, and my undergraduate thesis also had original results in graph theory, but I went into computer programming for a couple years and didn't publish before those results ended up (completely independently) as part of someone else's PhD dissertation.  If you want details, e-mail me; I'm using my real name and can be easily found by Google.

@CameronWilliams That sounds not only possible but also great! Could it be something like introducing topics in say advanced X to elementary X? Like introducing, maybe heuristically or maybe rigorously, some topic often skipped in elementary X (maybe calculus?) but discussed in advanced X (real analysis maybe?) ?

@JackBauer Sort of like that, but not quite. More like: "I'm studying representation theory of locally compact groups but I'm an undergrad and this is pretty hard so I need to boil it down in easy to dissolve chunks. Here's a unique way I found to explain/understand something that not many people have discussed." As an analogy, I did a project on Haar measures and when showing that they exist, there is one object that seems rather.. unmotivated.. called the Haar covering number. No one ever describes what it is. I thought about it and thought about it for a while and realized it was..

@JackBauer basically just a Riemann sum in some sense. In every proof it was just used as a tool without any justification or intuition. That's the kind of thing I'm talking about: giving a survey and then really deconstructing in ways that make sense to a novice.

Cameron Williams, given "pure mathematics operates at a level that is not very accessible for most undergraduates, even those doing research" can you help here How much knowledge is expected of a PhD applicant as compared to a postdoc or a research assistant? or here Emailing professors: Guidelines on choosing papers please?

The answers so far contain a yes and a no, so let me add a yes-and-no.
Undergraduates can - and often do - prove new things, but hardly ever anything of importance. It is up to the advisor to find an interesting question which is simple enough to serve as the topic of a thesis, but not yet dealt with in the literature. Different from a Ph.D., a bachelor or master thesis is heavily constrained in time, so as an advisor you should only give a topic if you are pretty certain that something can be done by an unexperienced researcher in short time. On the other hand just repeating the literature is boring for the student. One way to find good topics is to look at what is often referred to as folklore: Every textbook contains the theorem that X implies Y, and every expert knows that quasi-X already suffices, but noone bothered to write it up. This will most probably not be worth a publication, but proving a theorem not yet contained in the literature is motivating. Another simple method is looking at all the things you excluded from your own papers. If you worked out an example, but did not include it in a publication, you can let the student generalize it.
What you should not do is ask a student a problem you are really interested in. First the student will be frustrated, because the problem is too hard for him, then you will be frustrated, because you will spend much more time explaining things to him then you would need to find the results for yourself, and finally everyone is frustrated, because you find an answer and have to explain it to the student.

"Every textbook contains the theorem that X implies Y, and every expert knows that quasi-X already suffices, but noone bothered to write it up." Are there a lot of things like that just lying around? For example?

If someone proves a result, which only serves as a tool, the conditions are quite often too restrictive. For example, Hilbert space is used where reflexive Banach space suffices, or compact can often be replaced by countably compact. In number theory you can look at older paper using  exponential sums, and see what improvements for the latter yield in the application.

Jan-Christoph Schlage-Puchta, "the conditions are quite often too restrictive", do you mean it wouldn't be of interest to many mathematicians anyway?

When talking about topics for a Bachelor or Master thesis, I think about problems which are open in the sense that they are not published, but solved in the sense that every expert in the area could immediately write down a proof. So I don't think these questions are interesting to other people.

@JackBauer Most of the work is in figuring out what these things are.  If I had an example which I knew well enough to cite it here, probably somebody would have proved it.

Yes, undergraduates frequently prove new things, in the sense that every year there are new, publishable results proved by undergraduates. So, although a relatively small number of undergraduate math students participate in true "research", there are certainly students who are able to make nontrivial discoveries as undergraduates, and more than one might initially think. I have been at prestigious research schools and at anti-prestigious regional universites in the U.S.A. At every school I have been, there were undergraduates in mathematics with the aptitude for publishable research. The talent needed may not be "common", but it is certainly not "rare". The obstacles are primarily cultural, not intellectual.
The topic of undergraduate research has also been the subject of a question on MathOverflow, which makes for good reading.
For an example from personal experience: I recently published a peer-reviewed paper in what I consider to be a high-quality journal (and which is not in any way a "student" journal), with an undergraduate student co-author, who discovered the proof of one of the main theorems on his own between two of our research meetings.
Another example is the journal Involve, which is devoted to genuine student research. From their self-description:

Involve showcases and encourages high-quality mathematical research involving students from all academic levels. The editorial board consists of mathematical scientists committed to nurturing student participation in research.
Submissions in all mathematical areas are encouraged. All manuscripts accepted for publication in Involve are considered publishable in quality journals in their respective fields, and include a minimum of one-third student authorship. Submissions should include substantial faculty input; faculty co-authorship is strongly encouraged. In most cases, the submission (and accompanying cover letter) should come from a faculty member.
Involve, bridging the gap between the extremes of purely undergraduate-research journals and mainstream research journals, provides a venue to mathematicians wishing to encourage the creative involvement of students.

One thing that undergraduates are unlikely to have is the breadth of knowledge that is expected for PhD recipients. Particularly in mathematics, PhD students are examined in a range of subjects, and are expected to have mastered large parts of the undergraduate curriculum.  Undergraduate research often involves learning enough about one particular area to prove new theorems. The student still needs to spend time learning other areas to have the knowledge expected of a PhD.
The real key for undergraduates who are looking to do publishable research is to find a collaboration with a good faculty mentor. Independent research by undergraduates is indeed quite rare (in fact, the majority of mathematics papers currently published have two or more authors - even experts benefit from collaboration). The MathOverflow thread linked above has more advice from other mathematicians.

Thanks Oswald. Your example is kind of strange. WOuld your undergraduate co-author even have the opportunity to do such if not for knowing you?

Perhaps I should have said: I heard a PhD is like an original contribution or something. Doesn't proving something new kind of amount to an original contribution? Again, I understand this may seem stupid.

I think it would be much harder for an undergraduate to publish a paper without any interaction with faculty - but it would be hard for anyone unfamiliar with the customs of mathematical publication to write a paper suitable for publication, even if they had great mathematical aptitude. // I agree that proving something new is an original contribution, and a PhD thesis is normally required to have enough original contributions.  There is not that much difference between research for a PhD dissertation and research in general, though.

I wouldn't say undergraduates frequently prove new things, especially not in pure math. A small number of math undergraduates do serious research and even fewer make major contributions to the work. Most undergraduates hardly have the mathematical chops and insight to make major contributions simply due to lack of enough exposure.

I think we are answering different questions - I am answering "how often is new research produced by undergraduates", while I think you are answering "what is the chance that a random undergraduate will produce new research". @Cameron Williams

I respectfully disagree. I'm saying that you're way over-inflating how successful undergraduate students are and my guess is that it's because you've worked with some very successful ones. My point is that on average, so very few that actually do research make contributions. Hell, successful PhD students maybe end up with only one or two papers by the time they're finished.

So far, I've worked with three (sets of) students at a non-selective school, resulting in three peer-reviewed papers that, in their journals, are indistinguishable from any other research. The students all met the usual standards for co-authorship. (This record is partially because, as a researcher, I know enough to pick math problems where we are likely to find publishable contributions.) When I was at prestigious research schools, I saw even more math majors who would have been able to work on publishable research as undergrads. As I wrote, the issue is much more culture than aptitude.

I am not sure this is an answer to the main question asked (meaning the titular question, not the final line of the post), which is about most undergraduate theses, not things like REUs.  So the question is not just about what students are capable of, but what their undergrad thesis advisors think is a good project.  I often think it is more beneficial for a student to learn some important established math than trying to do something new which requires little background.

@Kimball: I think that is one of the reasons for the vastly different answers.  My answer is intended to respond to the final question, not to the title of the post, which asks something quite different. My reading of the post overall is that it is progressing toward the final question, asking whether undergraduates are even capable to proving new results.

One example I know of an undergraduate math thesis that proved something new is Manjul Bhargava's 1996 Harvard thesis "On P-orderings and polynomial functions on arbitrary subsets of Dedekind-type rings", later leading to publications in the Amer. Math., Crelle's journal, and J. Number Theory. However this example really demonstrates the difficulty of undergraduates publishing anything new. Bhargava was not a typical undergraduate; he later became one of Princeton's youngest ever tenured professors and won a Fields Medal. It's not reasonable to base general expectations on exceptional cases.

I can tell you my experience as I am currently writing an undergraduate thesis (though as a summer project).
I am an undergraduate student in mathematics currently doing a summer « introduction to research » internship. I'm studying probability theory.
As a first year student, about half of my time was spent solidifying my mathematical background in probability, measure theory and analysis. I also spent quite a lot of time studying specialised articles, and finally I applied the general theory I studied to a specific problem, where I did prove something « new », while very closely following other published results. On the way there, I also proved a few lemmas, that, while not of general interest, are « new » and interesting to me.
Clearly, undergraduate students are not expected to find groundbreaking results of general interest. However, they can contribute to mathematics by summarising and gathering related results from multiple articles, applying new theories, finding examples, etc.
A word of advice.
You should not aim for great discoveries, but rather simply try to do your own mathematics. Ask yourself a lot of « stupid » questions and find their answers. That's how you'll end up with a few small new results. Make sure you can grasp the big picture of your field of study, that you look at it from a critical standpoint and that you understand the issues that motivate it.
Do math undergraduates frequently prove new things?
Yes. But not great things, and sometimes things that might already be known to experts (but not widely accessible). I think that it is good enough for an undergrad to prove things that are new to him/her and her classmates/advisor/etc.

Some of the implicit premises of these sorts of questions, or the implicit premises in responses to the question, are really the issue. I would heartily agree that undergrads of all "calibers" should "be in the room" when something resembling "live" mathematics is being discussed. But/and this is most meaningful when we look at the falseness, artificiality, and sterility of the typical undergrad curriculum: it's fake and moribund, with no immediate room for anyone to do anything at all, and no hints about reality, either. Ghastly, yes. But that does not immediately entail a sort of "opposite", that novices need know very little to make meaningful contributions. Raw cleverness has already been exercised, quite systematically, for some hundreds of years (thousands?). People have learned useful things, and to not know these is to not know how to change a tire, or a light bulb, or a furnace filter, or open the door. Not that the usual curriculum helps much, either, I agree! But that does not mean that basic operational skills (involving occasionally subtle mathematics, literally, here) are irrelevant. Getting outside the degenerate "school math" thang is excellent... but thinking that that means "we don't need to know anything!" is obviously silly... even if appealing. "Complicated".


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Contacting previous authors instead of contacting the associate editor/editor in chief
Is it okay to contact authors (in the same research area) who published in a journal to which I submitted lately, in order to get information about the peer-review process, or will this be considered unprofessional and one preferably should ask the editor in chief/ associate editor personally?

What information do you need about the peer review process that isn't available to you already e.g. in the instructions for authors?

I would only do this if I had an existing relationship with them, even just having met them at a conference. Otherwise it is a bit strange. The editor can probably help you understand things. There is also a canonical question here about journal process that you might find helpful:
What does the typical workflow of a journal look like? How should I interpret a particular submission status?

This is a topic one often discusses at coffee meetings with colleagues at conferences, so it's a fair question to ask others. I see no reason why you shouldn't be able to ask it of colleagues. Of course, if I got an email from someone I've never heard of about this, I'd probably think that is an odd request.


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likelihood of Master's application being rejected due to obsolete GRE and IELTS scores
For those who submit obsolete GRE and IELTS scores (good scores), how likely is it for the admissions committee to reject/throw away the applications due to this? Will they instead review applications holistically?

This is relevant : https://academia.stackexchange.com/q/122420/72855

GRE/IELTS scores are far from the most important part of your application. The faculty reviewing your application are likely to overlook this if the rest of your application is strong.
However, the problem is that the applications may be filtered by computers or administrators before they reach the faculty (e.g., to remove incomplete applications) and they tend to be less forgiving.
A particular concern for you is that your scores will need to be verified at some point before you can register as a student. This requires you to get ETS to send the scores directly to the university. However, as far as I know, ETS will not send expired scores. Thus you will not be able to verify the expired scores and you will not be able register.
As annoying as these tests are, I suggest you re-take them to avoid the possibility of missing out on grad school for a silly reason like this. If you really cannot re-take the tests, you should try your luck, but I suggest that you make it clear on your application that the scores are expired. The worst thing would be for the university to think you have misled them.

This really depends on a million other things. If you're an outstanding applicant in every other way and this is the only issue then probably not. If it's a choice between you and candidate X who has everything in order, then yes - this would be an issue.
If you're applying in STEM (as opposed to, say, English literature), then this may be less of an issue: the school may make you take some internal English course to make up.
In any case, please do not lie in your application: there's no issue with submitting an old IELTS, but there is an issue with making it look like you just took it.


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How relevant is it, if updates to a journal paper are not reflected on third party websites?
My friend published a paper. After few months he found some error present. He asked the journal editor for a correction, made the changes and replaced the old version with the new one. The old journal paper is still present in websites other than the journal. Will these websites replace the old with the corrected new paper? If not, will it be a problem?

Actually, I'd worry if others are republishing a paper from the journal at all. Was it done properly?

It depends on the third party website. If they are constantly monitoring the main journal, it might happen, but I would assume that this is not true for every website. On the other hand, in most cases the journal should only be available via the journal web site thanks to the copyright regulations of that journal. And people citing the work should always look for the original source.
Therefore I would say it is not a problem of your friend, maybe it can be a problem for people working with unreliable sources. But I would not care.

This is why journals don't like to make corrections to published papers. The journal can fix the version on their website, but once electronic versions of the paper have been sent out, they can't be recalled, and it's beyond the journal's control to replace all of them.
As OBu said, it's up to the third party websites to replace the old paper with the new one, and this depends too much on the website to predict. As for whether it'll be a problem, I'd guess no: if the error is major enough that it'll be a problem, the journal should absolutely not change the paper but should issue a corrigendum. Since that didn't happen my guess is that the error is too minor to worry about.