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In physics, does the order of authors on a paper matter?
I'm writing to inquire whether or not the order in which the names of the authors is mentioned has any bearing? Here's my case:
I started working on a project with a professor in the physics department last summer. The project is almost finalized and we're in the stage of finalizing the final draft for submission. Since I'm an undergraduate, I'm not the one who came up with the problem. The instructor suggested the problem to me (and now that I think of it, the problem seems like a very natural question to ask based on a work the said professor published last summer). He has been available for feedback all throughout the duration of the project. For instance, he helped me derive the first closed form result of the paper; he also guided me through every stage of the project -- from guiding how to approach to the problem to helping debug the code when I couldn't make it work in the initial stages.
Having said that, the professor didn't contribute directly to the project, in addition to helping and guiding me throughout the project. I'm the one who worked through most of the results and went through all the computational tasks. In addition, I'm the one has written the final draft that we have uptil now; he simply helped me edit the template once.
When I met the professor the last time around, he was editing the template and he listed his name as the first name, and my name as the second name on the paper. Naturally, I didn't say anything.
My question is: does the order in which the names are mentioned matter? Does the answer to this question vary from discipline to discipline, even from one area of research to another area of research within the same discipline.
Here's some information that may help answer the question:
The nature of the physics-related paper can be classified using the keywords: "Quantum Dynamics," "Quantum Mechnaics," "Quantum Information" and "Open Quantum Systems." We're trying to get the paper published in Scientific Reports.
Side note: I'm an undergraduate student. Since I'm not technically affiliated with a university (I pay to attend the university, the professor is paid to be there), does this have any bearing on the answer to the question.

See also What does first authorship really mean?. Since this related question does not have a physics-specific answer yet, I hope it's OK if I change your question to be about physics specifically.

If you are a student, you are affiliated.

Are there only two people on the paper? If only two, don't worry that much either way.

@AJK Yes, there are only two authors. Should I be worrying too much about it? The only thing I'm concerned about is how would this issue affect how admissions committee (graduate school) view my potential publication.

@JunaidAftab Most students (at least in math) who get accepted even in top schools' graduate programs haven't published anything. So do not worry.

Physics has two broad conventions: the closer you are to high energy physics/particle physics/string theory/mathematical physics, the more likely you are to see alphabetical ordering. However, within condensed matter and biophysics, you generally see "First author = trainee primarily responsible for work." (I think astrophysics is more mixed).
There is actually some empirical data on alphabetical authorship by field here: Waltman arxiv:1206.4863 which confirms this general impression.
In quantum information, you might be at the edge of the condensed matter and "mathematical physics" conventions. Take a look at your advisor's other papers - are they alphabetical? Or are grad students and postdocs within the group usually listed as first?
Also, within fields where author order matters, the implication of "second of two authors" can be very different than, say, "second of five authors." With two authors, there is often the presumption of nearly-equal contributions. However, someone who is second of many may have, e.g., provided a figure's worth of data.
Your description would be broadly consistent with being a first author in most fields. (This assumes that there are not other people who contributed, who you have not mentioned - sometimes undergraduates are not given the best picture of what contributions are there.)
If this does not clear things up, I think it is fine to ask your advisor why he set the author order the way he did.

The professor hasn't written papers with students -- undergraduates or postgraduates -- before. I'm from a college in Pakistan; over here, the research culture, for one, is pretty bleak. Moreover, the we don't have a lot of postgrad's here. The professor has almost exclusively collaborated with his PhD supervisor; I'm not too sure whether the ordering was alphabetical in those cases since my professor's name starts with an A. I don't really understand the second last paragraph. Also, I'm still confused what to do? Any suggestions?

The second-to-last paragraph is saying that in many fields, you would be chosen as the first author, unless there were other people who did more work you didn't mention. (Not applicable, as I see there is only you and your professor!)

My advice is that first or last authorship on a two-author paper will probably look equally as good in grad school admissions. In your CV or other application materials, you can briefly describe the work you did (and hopefully your advisor will also note this in your letter) - this will show what you are capable of. You can even note that the authorship was arranged alphabetically, to clarify this point.

There is also probably nothing wrong with asking your advisor why he changed the name order, as long as you do it politely, e.g. "I'm new to the field and trying to understand its conventions - why did you choose the names to go in this order?" This will probably not change anything, but will help you understand what your advisor is thinking.

(Definitely be as polite as possible - this is the sort of thing advisors can be touchy about, and if you are already feeling hurt, you can end up asking these questions rudely by accident.)

In physics, typically the first author is the person who did the most work.  The last author is the person who supervised the work.  If there are multiple supervisors their names go near the end of the list.
There are exceptions, particularly in cases where the number of authors is very large.

I am not sure this is universal -- for example, I thought that in high energy theory it is almost always alphabetical.


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How to best utilize a gap year to enter mathematics graduate school
I have the following two options available on how to utilize the next year. First some background:
I'm an Economics and Mathematics major at a college in Pakistan. I'm, however, interested in going to graduate school in mathematics and working with mathematical physicists. Starting from the end of my second year, I started taking physics course. I'll have a dozen or so physics courses by the time I graduate; I have the basic, and a few upper division math courses. I continue, however, to study mathematics outside of class.
First Option:
I can choose to take an additional year before going to graduate school. I doubt I'll be able to fulfill the requirements of a major in mathematics (and my school's policies are a bit ambiguous as well); however, at the very least, I could take as much mathematics courses as I can. If I were to do so, I'll probably be able to complete the department's requirements of math courses, so one could assume that I have completed all the requirements necessary to fulfill the department's choice. I may be offered a RA position in the physics department; so I could fund my additional year with that money, along with all my savings.
Second Option
Just take the year off. Continue to study mathematics informally with the instructors. That is, cover material outside of class/college, follow up with them regualrly so they can write good letters. Depending on whether I get to work on some other problem, I could still take the RA position and save money to apply for graduate schools etc.
Here are some more details that'll help you advise.
Mathematics courses I have taken:
I have taken single variable calculus, multivariable calculus, linear algebra, real analysis (two semesters), probability and point-set topology.
I have also been studying material outside of class. I finished linear algebra II on my own, and I am working on multivariable analysis right now. Regardless of which option I choose, I will continue studying more mathematics, revising old mathematics etc.
Here's my assessment of the options thus far.
RA Position:

Frankly, I don't think I'll want to work in this position for an extended period of time. I wrote a physics paper with a professor on (open) quantum dynamics; since he wishes to work on similar problems, he offered me a position so I can continue working on similar problems.
I don't think it'll be (extremely) helpful in my eventual application because the path I wish to take is somewhat different.
I, however, can get paid and use this money for either option.

First Option

I get to cover my cores -- ODEs, Complex Variables etc.
I'm not too sure how much it'll help, but I'll have graded coursework.
I'll still, however,  be covering some of the same material outside of class on my own.
It'll be hectic; I'll have to complete 2 semesters worth 35-40 credit hours. I will also have the responsibilities of the RA position; on top of that, add applying to graduate schools in the mix.

Second Option

Flexible schedule designed at my own will and pace.
If both the above options would have the same effect, then I can just study material on my own. For instance, assuming I take the RA position, I'll be on campus the next year. I can audit courses which my advisers will be teaching next semester -- for instance, on of my advisers will be teaching a graduate level course in algebra. I can sit through classes, work on problem sets, and hopefully then the instructor can compare my work when she writes the letter; of course, I won't be graded relative to my peers, though.
Another local university provides short, intensive courses in mathematics. It's a graduate school, so most courses are advances courses -- for example, in advanced algebra, differential geometry. Since I'll have more time on me, I can perhaps take 1 or 2 of these courses, and cover the more advanced material. You can see the this link for a description of such short courses that have been offered in the past.
Also, with the first option, I'd have to go through the boring, plug and chug courses -- ODEs etc. Very important to train yourself, though.

It'd be great if someone could weigh out both these options, especially considering the graduate admissions process at math departments.

I think whatever you do, you need to study a lot more math to get into math graduate school, and you need to study it in a way such that an admissions committee can verify it. Because your description of the math courses you have taken so far would be way below what is expected from someone wishing to major in mathematics.

@TobiasKildetoft True, how do you see such programs, for instance, though: http://dept.math.lsa.umich.edu/graduate/admission.html#GenReq. This department lists that it isn't necessary for one to be a math major. See the stated requirements. Also, what's the best way to make sure admissions committees can verify what I have done. Hence the question.

For secondary reading (not primary because they should not replace well known standard texts), I recommend getting copies of All the Mathematics You Missed: But Need to Know for Graduate School and Some Modern Mathematics for Physicists and Other Outsiders Volume 1: An Introduction to Algebra, Topology, and Functional Analysis and Volume 2: Functional Analysis with Applications.

See also the suggestions given for the StackExchange question Self-study mathematics subject sequence and recommended books.

What country are you thinking of applying to? The admission process in Pakistan is potentially very different from the US, UK, India, ...

@StrongBad Depends on the funding, but the US, Canada and if funding is available, I'll look into European universities as well.

What's the best way to make sure admissions committees can verify what I have done?

Your transcript and your GRE score.
No one ever became a good mathematician without doing plenty of self-study.  But it is hard to get admitted in the U.S. as an international student without a strong transcript.  So, if the question is, take courses or study on my own, the answer is both.
It sounds like you find the idea of sitting in a cave learning math on your own quite appealing (which is understandable).  Well, this is your decision and I doubt I'll have much influence, but I will ask you to imagine yourself in sitting on the admissions committee looking at Junaid's application.  From the committee's point of view, Junaid who has spent the last four to six months doing self-study might not be as good a bet as Junaid who has been continuing to work his way through the available math courses in his local university.
Are there some periods, perhaps a summer break, perhaps a couple of weeks here and there, when you can do some cave visits, without setting yourself up there for the entire duration?
Why are you so down on ODEs?  In a classic ODEs course, you lay groundwork (e.g. Laplace transforms) for some great stuff to come (e.g. Fourier transforms -- so useful in physics).  Are you basing your negative remarks on things you've heard from students who already took ODEs in your university, or what?
Apparently you haven't run out of courses to take yet in your local university.  This is good news.
I can't possibly weigh in on the research assistantship and the financial question, because I have no idea what the alternatives are, or what your general financial situation is.
Having followed your questions here for a while, I am glad to see from your comment that you have taken a close look at at least one math department's admission application guidance.  Good work!  Are you planning to take a first look at a math GRE practice exam some time soon?  Are you looking for a mentor who can help you structure your GRE preparation?  Have you started to look at some faculty bios in a couple of departments, to identify some departments that have several people doing the kind of work that appeals to you?
You give the impression that you have done good work so far, you are a hard worker, you are motivated, you enjoy studying math, you have found good study techniques, and you have a vision (a dream).  Now add a soupçon of patience with yourself, and you'll be all set.

I didn't see anything in the question about being US admissions.

@StrongBad - It was in previous questions by OP, e.g. https://academia.stackexchange.com/q/86933/32436.  But I see Junaid is now thinking of looking more broadly.  I hope he will prune his tree of options realistically, and eliminate countries that would require learning a brand new language (he speaks English quite well, judging from his participation here), and those that don't provide a stipend.  My understanding is that he needs full financial support. // For which countries would "it is hard to get admitted as an international student without a strong transcript" not be true?

I like option 1 (do an extra year of school) better.  It gives you some organized support for your study (most people lack the discipline to really do that for themselves effectively).  Also it proves the work objectively to an applications committee.
Yes, the RA will cause a conflict but at least it keeps you in touch with physics.  You also need to talk to the person providing the funding and get some clarity on how many hours he expects or conversely how many hours you will need to study math.  Just man up and talk to him about that.  This actually applies in both options, but the self study option has the disadvantage of the professor tacitly assuming you are dedicated to his RA and not honoring your self study time the way he would course work.
It would have been helpful if you also listed what coursework you did in physics.  I'm baffled how you can do several physics courses without ODE for instance.
I'm also confused with how/why you took all the physics courses and didn't get your math (or econ!) degrees done.  Nobody is perfect and the situation is the situation but I just wonder if there is some learning here.  At least you appear to be more strategic now.
It would also be helpful to know status of your econ major.
And then is some sort of BS going to happen at the end of this year or is it just a case of you played with 3 subjects and never got enough done to get a degree in anything?  If so, this is another strong rationale for doing the extra year of coursework (however funded) and just getting the darned sheepskin.
I agree that your math background is insufficient for math grad school.  Probably also for physics grad school.  A semester of ODE is expected and usually an extra 1-2 semesters of "engineering mathematics" (a brief coverage of linear algebra, PDEs, and complex analysis, with emphasis on key calculation methods and problem types, not proofs).
If anything maybe your background would be a good fit for econ grad school in the states.  Econ tends to be more mathematical in grad school than undergrad (which frustrates many undergrads who love micro but hate derivatives).  You would have better than average math background in econ but much worse than normal in math grad school and somewhat worse than norm in theoretical physics.
I also suspect that your math ability (intrinsic ability to learn math) somewhat correlates to your education level.  If you head to econ, you will have a comparative advantage.  If you head to math or physics, the opposite.  Also, econ grad school has traditionally been somewhat receptive to people moving cross field (e.g. from engineering undergrad to econ grad school).

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Contacting professors via email
I am an international, senior Economics and Mathematics major at middle-sized, private international university. I'm trying to finalize the list of schools to which I will apply (to Math PhD programs in Mathematical Physics), but I am finding it very difficult to contact professors given my unconventional background.
I know that I am not a candidate for admission at top-tier universities. For one, my major isn't either mathematics or physics, even though I have been taking physics and math courses for the last 2 years. Also, I'm sure the professors who will write my recommendation letters aren't very well-known either. I may get to work with a professor who has a good reputation in the US, though. Though this depends if the professor accepts me under his tutelage, since he left my current college some time ago. By the time I graduate, I would have taken a dozen physics courses; I'll be lighter on the mathematics side, though.
I'm not sure if I qualify to be considered as a candidate in the mathematics departments; I want my training to be like a mathematician, but I'd like to work on mathematical physics problems. So, I'm not sure how to gauge how fit I am for a department? To that end, I'd ideally like to email some professors, explaining my unconventional background, express interest in working under them, and get advice if I will be considered in their program.
What's the best way to structure such an email? Should I give an extended summary of my "situation"? Is it common to elicit "advice" from professors; we all know they don't have much time on their hands; do I need to read their papers to contact them before hand? Etc.

Did you intend to link a question in your first sentence?

Junaid, I am not sure you have completely processed comments and answers you have already received to previous questions.  Look, it's really pretty simple.  Pick some programs you're interested in and some researchers you're interested in.  Pick at least one "safety" school (that you think you would be 90% certain to be accepted to), one medium and one "stretch" school (that you think it would be a stretch to be accepted into).  Start preparing your applications.  If you want to write to a specific researcher (and please remember, in the U.S. an individual researcher would not be making...

admissions decision), go ahead.  Keep the email short.  Attach an unofficial transcript and a CV.  If you don't have an unofficial transcript in electronic form, just list the courses you've taken.  Explain in one paragraph what you'd like to study.  The application is where you'll go into more detail.  Then pose some specific questions, such as "Should I apply to the math department or the physics department at your institution?"  The worst that can happen is that you don't receive an answer.  Worse things have happened.

Before beginning to email academics at other institutions, I would recommend talking to your tutor/ advisor/ referees first, to find out what route they think is best for you to take. It may well be that they know of certain professors or departments that would be a good fit for you. It would also help your letter writers to understand your goals and hence write a better reference.
Having said that, emailing other academics is a good idea. I have done something similar in the past, with very good results (the advice I received from one professor in particular directly led to me finding a supervisor for my Masters and an offer for a PhD place).
The best thing to do when writing such emails is to keep it short and simple. You are already aware of the time constraints professors have, and an overly long email may be off-putting. Aim to summarise your situation in one or two sentences and emphasise your interest in their field of research (looking carefully at a few of their most recent papers before clicking 'send' is a must).
The resulting email may look something like this:

Dear Professor X,
I am a senior Economics and Maths major at Y University. I have found over the past two years of taking physics and maths courses that my real interest lies in mathematical physics and I would like to do a PhD in this area, possibly at your university. I noticed that you have a recent paper on Z theory, which is something that really interests me. Is it related to the P theorem from this earlier paper of yours?
Due to my non-mathematical background, I am unsure whether I should apply to the maths or physics departments at your university. Can you advise me as to which would be the better fit for my skills and interests?
Many thanks and best wishes,
Junaid Aftab

Be prepared for some people to ignore your email entirely, or to reply with a very short response (e.g. "Thanks for your email, unfortunately I can't help you with this matter at the moment. Best of luck."). The key is to be persistent. Find someone else to contact. Keep trying.
If you are successful and someone replies with helpful advice, ask if you can follow up with a chat over Skype. I have always found face-to-face conversation more effective than email, and if you are seriously considering doing a PhD under that professor it doesn't hurt to get to know them a little better and get a feel for what they might be like to work with.
Above all, remember that quite a lot of people in senior levels in mathematical physics are interested in mathematical physics and are glad to find interest in mathematical physics in other people.

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How to encourage and motivate undergraduates to regularly and independently work on problem sets?
I'm wondering how can a professor encourage students to complete their problem sets on their own in undergraduate classes. I'm not saying that every student cheats off of other students; rather, I'm saying that some, if not most, students free ride and copy the solutions from their peers (or from the Internet). In addition from my experience, I feel as if this problem is more pervasive among the non-science majors/students (at least science majors are given assignments; the number and quality of them and the level of studens' interest to get on with them are spectate issues). I've even heard a professor remark: "since I know most of the uninterested students in the class are going to simply copy off solutions, I won't be giving assignments; the serious students amongst you (who'll go to graduate school) can do assignments over there." Frankly, I was appalled upon hearing this statement.
I come from a college/place where the local cultural norms are such that students don't like to sit down and devote week in and week out on problem sets, and the professors (for the most part) don't make a serious attempt to give problem sets that really consume a student's week or so. I'm sure the professor who'll dare to give a problem set every 3 lectures (as is the norm in most U.S colleges) will be frowned upon by most students.

Given such a context, I am interested to know how can professors, in the larger scheme of things, encourage their students to independently solve problem sets on their own? Also, are there any good strategies that professors can use to make problem sets which the students would eagerly wait to solve week in and week out. I know one can only change the mind set of students from the ground up. But how can professors teaching at the tertiary level cope with such (deeply rooted) problems?

Thanks.

Do the faculty and administration generally care how well students are actually educated, or do they have other priorities?

Among the faculty: some of them actually care. But even among them, I feel as if they have gotten demotivated with time; it seems as if they know that the students aren't going to work that much, and this inevitably results in giving them less work. Ideally, I'd want to see professors do the opposite: make the classes harder, give more sets, and encourage the students. The question is: is that possible/manageable or am I being too optimistic? Honestly, I at times don't like the academic environment of my college.

My question is higher order.  Let's say a bunch of professors started demanding more of their students and enforced these standards.  Naturally, fewer students will pass, and some of these students will complain.  Will the end result be that the students are told that university should be hard, or will the end result be that the professors are told to make their classes easier?  (Note I don't know - and it doesn't matter for this question - whom the students complain to or how this final result is determined.)

A few (by no means exhaustive) suggestions:

Use extended-response or open-answer style questions. What such a question would look like is completely field-specific, but it's definitely doable in every field (yes, even the technical ones). In theory at least, this makes it harder for several students to copy off of another student without commiting actual plagiarism.
Avoid assigning problems directly from a textbook. This requires more effort on your end, but it prevents students from being able to just copy out of a solution manual.
Make the students propose the problems. Similar to my first suggestion -- if the problem is that students are uninterested, give them the chance to do something that is interesting for them. Obviously you need to put bounds on this, but even within the confines of a single problem set this is doable. For example, in a course on Data Structures in CS, you could ask something akin to "write a simple program that uses data structure X and methods A, B, and C." What they do is up to them, but you still test their comprehension.
Use a flipped classroom model. This is probably the most involved suggestion, but if done properly it means the students are working on homework right in front of you. Makes it a lot harder to just copy answers when the instructor is walking around the room.

Here's the horror story (which does happen) I'm worried about: 1) Professor uses flipped classroom model.  2) Students show up to class without any signs of having prepared, either because they are lazy or illiterate.  3) Students do no work in class unless professor is standing next to them and telling them word-by-word what to write down to do the problem.  4) Students complain to administration that professor is not teaching them, since, in their view, teaching is equal to giving step-by-step instructions for everything so students don't have to do any thinking. (cont)

Administration, desperate to retain students due to financial troubles, fires professor.

@AlexanderWoo: Funny enough, (1) and (2) are almost exactly what happened with the only flipped classroom I was a student in. The professor posted videos online for us to watch, and few of us did (I'll admit, I did not). However, (3) and onward did not happen, as once the professor realized what was happening, we started getting daily quizzes on the material (before working on the homework).

As an undergraduate student, here are some things that motivate me to complete problem sets that are regularly given out:

Have the problems be interesting to think about, and lean towards difficult problem sets due less often rather than easy busywork due every class.
Encourage collaboration, while having harsh penalties for copying. The students should be discussing your problems, if they're interesting and enjoyable. This isn't always a bad thing!
If you're really worried about cheating, have a problem set that they are not allowed to discuss or do interview grading - 10 minutes talking with the students will allow you to tell if they understand the material.
Use problems that would come up in the real world, that are related to what students may want to do in the future.

These are a few things that I've had professors do that I liked, that encouraged me to work hard on the problem sets. Of course, there will be cheating, but if students are interested in your subject, these are ways to encourage that interest.
(Note: I am studying computer science, so these answers may not apply to your subject.)

Customization

make the parameters of the problem specific to each student. Customization is the only way that keeps students from being able to just copy and paste the solution from each other. It also still keeps them engaged as they have to tailor the solution they might find available to them.

Flip the Classroom

Make them create some questions and answers, and then other students have to solve it. This way, they can not come up with some questions or answers.  randomize the allocation every week. grading can also be allocated the same way, but normalization of grades might be an issue depending on difficulty of questions.


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Online vs hard copy, which texts are students more likely to read?
I am wondering if anyone knows of any studies that indicate whether students read online texts more or less than hard copy texts.
I am in the midst of trying to "reboot" the intro stats class I teach and a big question in my mind is how effective the text is. Active learning (like the flipped classroom) requires the students read the material beforehand, and I'm not convinced (from the results I saw this year using a Pearson Learning Management System (LMS)) that the students are reading the online text very much.
I have a suspicion that students read the online texts less (and I plan to give my students a survey to check) but I'd like to know if there is any research supporting my limited observations.

Studies have shown that students don't read hard copy textbooks either- unless you make it necessary for the students to read the text to get points on quizzes or something similar.

@BrianBorchers: Could you reference these studies?

My Intro to Quantum Mechanics professor combated this problem by not giving us a practice exam or telling us anything about the exams. He said that he wanted us to know the material well enough that we would know the important topics in the class, which meant getting to know the textbook inside and out. While I hated it at the time, it actually forced me to learn the material so I could pass the class, instead of the "memorize the practice exam" method.

What is "Pearson LMS"? Yes I can search for it, but I think it is best to give an hint to whomever is not from... wherever it is common. Also maybe it is not common anywhere in the near future, while the general question may still be a good one then.

Sorry..."LMS" just stands for "learning management system". The term itself refers to a self contained learning environment (i.e. it provides the text, problems for online assignments, and also grades them). Many large companies with traditional roots in education (Pearson, MacMillan, Cengage) have branched out in this direction and provide such platforms that sync to course management systems like BlackBoard, Canvas, Sakai, etc.

This recent literature review goes over some of the research on print versus screen reading.
It looks like in real-world situations there are no significant differences in grades between e-textbooks and print textbooks even though some laboratory studies show that content reading works better from print and that some students tend to prefer print.
In contrast, course reserve books are more likely to be accessed if they're e-books rather than print books.
Practically, it may be worthwhile to give students a "tour" of the ebook to make sure they know how to use it, and/or to design quizzes or activities that require students to know/use the textbook material.

+1 for a great answer. Just a note: SE prefers links to be “hidden” or placed as footnotes using Markdown link text, since it makes the answers easier to read.

I will say personally that using both platform is far the best. Online text reading can be a little bit tedious especially when it is in large PDF volume for example and sometimes calculations may not be properly formulated especially for student new in statistics. From my experience having to take bio-statistics, i still had to refer to my  hard text book especially to see exactly how the answers were got from start to finish of the problem. On finding a report to support you. i came across the site below, check it out and see if the data can support your report.
http://hechingerreport.org/textbook-dilemma-digital-paper/

I agree that if it is possible, give students option to use either modality, but as a blind student, I would love online materials. We need to think about accessibility of our materials for everyone.

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How do I show others what I am capable of?
I just turned sixteen this January, and I have started thinking about where I will go to pursue research in theoretical physics and mathematics. I live in India, and, sadly speaking, research here in India is awful, even though many claims of scientific advancements in India are made. First, I would like to share what I do at home.
I have mastered differential and integral calculus, and I have independently proved, and discovered some of my own proofs of theorems in Real Analysis(Proofs which I haven't seen in some other source, but they might exist). I have also independently discovered the well known techniques of solutions to certain kinds of differential equations, without having any knowledge of how to solve them. I continue to explore higher level mathematics, and try my best to prove each theorem I encounter myself.
Now, the thing is, nobody at school, neither the teachers nor my friends, know anything about this. Even my parents are not fully aware about all this, but I try my best to share this with them. At school, the curriculum is such that the teachers are teaching to the test, and a heavy focus on contests and olympiads is laid, and to choose subjects of your own choice, you have to score high enough in subjects like Social Science and English, which don't even matter in research. Unnecessary debates and discussions are held, and I try my best to not share anything with these guys, to prevent getting into an hour long debate on what you should pursue.
In India, the major institutions are the Indian Institutes of Technology, Indian Institute of Science etc, where almost everybody focuses on getting into the IIT. To get an admission in the IIT, a standardised test, comprising of problems from Math, Physics, Chemistry and MAT has to be given. The admissions are solely decided by the score obtained in this test, which comprises of contest-like questions, which do not describe the research capability of the candidate. Even coaching centres have been set up all over the country, which train for this exam(called the JEE exam). The situation is really worse, and recently, I was demotivated by my teacher for not scoring enough in my social science exam. I have gotten used to this.
I would like to get admitted to institutions like MIT, Harvard, and I have read how I apply for them and what are the requirements for admission. The main thing is I have to distinguish myself from others, take part in extra-curricular activities etc, and maybe win math/science contests. But the thing is, I am not great at contests, which are timed, and I rarely would like to participate at school, where conditions are horrible. I have started taking leaves from school, because I think nothing productive is going on there.
Now, I would like to ask if I apply for these institutes, how do I share what I am capable of? As a young mathematician, I think I am really doing well, and will do well in research too. But, other than my notebook, where I write all my thoughts and all my findings, I don't have anything that can show my abilities. If I just tell my abilities to them, like I did here, they would not believe in me, which I understand.
So, my question is, how do I show the institutes my abilities? Is it necessary for me to win contests at national/international levels, which is not that easy?

You might want to consider less than stellar U.S. universities as well. There are probably over 30 U.S. universities at which you would not come close to reaching the upper limit of what they could offer you (courses including graduate level courses, classmates who will eventually get a Ph.D. in math or physics, faculty to work with, etc.), and if you really excel then you could attend graduate school at MIT, Harvard, etc., which would be more important for your future in mathematics or physics than what university you attended as an undergraduate.

You can try and get into the national team for the International Physics/Mathematics/Informatics Olympiad. A medal there can certainly get you far.

Could you demonstrate your talent by solving some important, open problem in physics or proving some important conjecture in math?

Take a look at these stories. Maybe there is one that could apply to you: http://www.gradschoolhub.com/10-youngest-people-ever-to-achieve-a-doctorate-degree/

@Justas Let's be realistic: He/She 's 16. Reading about open problems, if one can do it, is great. Expecting to solve one right away (or even before starting univeraity) is not. The sole case (in mathematics) that I have heard of is regarding an improvment of the bounds in the Bertrand's postulate. And even in that case, I am quite positive that there was guidence by a senior. @ OP As a side note, you can consider Grothendieck's story: Graduated from a far from bigname university (and even failed a year) only to revolutionize algebraic geometry later on.

If you are not truly a genius there is no way around standard procedures. But if you are:

Make a paper
Publish it in a journal
Get citations and acknowledgements if the work is good

That is how science works.
You can contact a professor closest to you for help. Show some raw paper made by you, and if you truly are capable then he will gladly help. The professor has a realistic set of journals in mind that might be willing to publish your paper. If you have published in some of the top journals, then you have earned a special way around standard procedures.
You do not need any highly branded college diplomas, recognition is all about skill. If you are capable to publish in a top journal without diploma, then I think that you have already accomplished your goal.

This answer is really appreciable. I always think about my work being my recognition, but, you know, psychologically I sometimes feel scared, and even when I am so confident about myself, I ask:"Will I be able to do it?What if not?". I know this is a stupid thought, which I should just throw in the trash, but when you constantly get demotivated by people around you, this thought may come. Anyways, I am really pumped up by your answer, +1.

@Siddhant JeffE is right, I cannot overemphasize that point. What you are asking for is special treatment beyond standard procedures for special treatment like getting in by success in competition. The only way is to show significant result. You should not expect any empathy from anyone, because you are trying to be on top of the top and not getting to top is already a non-empathy waking thing. As a person that got in with the success in competition I can tell that for many hearing about that was already kind of wow experience. This is a step beyond.

Many PhDs in math don't publish almost anything before graduation. It is unlikely that he has found something truly new, he just says that he has the potential and the ambition to study combined with talent.

To apply for admission as an undergraduate at ... (MIT, Harvard, etc.) ... their web sites shows the steps you need.  There is an exam (SAT, Scholastic Aptitude Test) such applicants may take.  Probably there are opportunities to take it in your country as well, but maybe just in one or two cities, so you may have to travel there to do it.  (16-year-olds generally haven't published papers.)


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Qualitative data but quantitative outcome; what type of research methodology is this?
Following are the steps, how I am working on a project:

Qualitative approaches to collect data
Forming criteria from collected data
Mapping criteria to function
Getting a matheimatical function to quantify data
Result in a (quantitative) tabular form.

Is this quantitative research methodology or mixed methods research methodology ?

Well, according to Alan Bryman in his research article Integrating quantitative and qualitative research: how is it done?, this is a theme that is still open to debate. He uses these 5 questions in separating qualitative, quantitative and mixed-methods approaches:

Are the quantitative and qualitative data collected simultaneously or sequentially? (Morgan, 1998; Morse, 1991).
Which has priority – the quantitative or the qualitative data? (Morgan, 1998; Morse, 1991).
What is the function of the integration – for example, triangulation, explanation, or exploration? (Creswell, 2003; Creswell
  et al., 2003; Greene et al., 1989).
At what stage(s) in the research process does multi-strategy research occur? (Tashakkori and Teddlie, 1998). It may be at stages of
  research question formulation, data collection, data analysis, or data
  interpret- ation.
Is there more than one data strand? (Tashakkori and Teddlie, 2003). With a multi-strand study, there is more than one research method and
  hence source of data. With a mono-strand study, there is one research
  method and hence one source of data. However, whether a mono-strand
  study can genuinely be regarded as a form of mixing methods is
  debatable.

In regards to your question, I would say that if by "criteria" you mean forming hypotheses that you later test with quantitative methods, that would clearly be a mixed-methods approach. If not, you are probably more in a grey area.


text:
Paper with potentially inappropriately-ordered authors, should a journal act?
A journal I handle recently received a paper that looks like a student's project. Reasons to think this include the research conducted, the writing style (e.g. the paper describes things which research-level readers are virtually certain to know well), and the fact that of the paper's two authors, one of them is a professor while the other's LinkedIn profile says they're an undergraduate.
If it is a student's project, then one would expect the student to be listed as the first author, but the professor is listed first. This feels a little concerning on an intuitive level (maybe bullying is involved?). All our communications address both authors, but the professor is the submitting author who also answers our questions, and the student has not told us anything directly.
I'm one of the journal's staff. Should I do anything? If yes, what?
Some of the obvious things I can do are:

Nothing. The "evidence" is at best suggestive and very far from conclusive. There could be nothing inappropriate going on at all. Besides, journals usually don't get involved in authorship disputes.
Directly bring it up with the authors. (The professor is likely to say the ordering is correct, but they were always going to say that, so this doesn't seem like it achieves anything.)
Email the student, and only the student, to ask them about this. (Could be scary for the student, might also offend the professor if they learn about it, which they easily could if the student tells them.)

Edit: to clarify, 1) there is no policy to email every author to get their consent to publish, but there is a policy to include every author in all our communications. I have no reason to think that the email address of the student is wrong. 2) The standard in the field is to list authors by contribution. For this particular manuscript as well, if the author order were alphabetical, the student would come first.

You are the the journal employee,  but don't the terms and conditions at journals usually specify that the journal has no say in author order? Wouldn't an intervention break this Ts&Cs?

@IanSudbery I don't understand. Journals don't usually arbitrate authorship disputes, but I don't see why we can't alert "someone" (maybe the head of department?) if we suspect bullying.

Is it your journal's policy to contact every author individually to confirm consent to publish, or to just assume the corresponding author speaks for them?  This case might be an argument in favor of the former...

Perhaps you should include the standard author-order format for the subject/field of the paper, since (as I've learned here the past few years) that standard could be for authors to be listed alphabetically by last/family name, that standard could be for the most-contributing author to be listed first, that standard could be for the most-senior author to be listed first, that standard could be for the most-contributing author to be listed last, that standard could be for the most-senior author to be listed last, etc. (Maybe some of these possibilities don't exist, I don't know.)

@DaveLRenfro I've edited that into the OP.

@NateEldredge I've edited that into the OP. We don't contact every author individually, but we cc every author whenever we send an email.

Is this the first time that this professor has submitted a paper like this? Or have they been doing this regularly? A one-off event will likely be handled differently than a long-term pattern.

Think scientifically! What can you support by evidence? what is just a notion? You may be correct but this is a the flaw of systems that depend on ethical behaviour while there will always be those who break it. In th eend it is not your task to try to second guess what is going on, it will be up to the authors to sort that out. This will occassionally fail but can bite the violator in th end in the long run.

I would strongly encourage you to go for Option (1) - do nothing.
You are reading a lot into a very noisy signal. It's possible that something shady is going on, but it is also very possible that the professor did in fact do most of the intellectual work and just got some help from their students (e.g., as part of a research internship). Contrary to popular belief, some professors do in fact still do research on their own. Do not apply your own experience from how things work around you to everybody in the world (remember that academia varies more than you think).
I would argue that with Option (3) you are severely overstepping your boundaries. Nothing good will come out of that. Even if there is something shady going on, what will your email accomplish? The students are already aware of the author order (they receive an email after all, and likely need to confirm their co-authorship). And if there is a brewing conflict because the students are unhappy with the order (which, again, is a pretty substantial if), there is nothing you can do to help them. You cannot force them to change the order of authors based on a hunch. The most severe thing you can do on the journal side is desk-reject the paper, but then the authors can just re-submit elsewhere.
I have multiple papers where I, as the most senior person on the team, did most of the work and was listed first. I would have been extremely miffed if somebody from the journal staff would have asked me to explain myself (or worse, show proof that I did indeed do sufficient work to be first author).

I upvoted, but a question about this: The most severe thing you can do on the journal side is desk-reject the paper - if there is something shady going on (which I agree is a big if), presumably I could alert someone at the university?

@Allure I mean, nobody stops you from doing that, but again - if you have nothing more than a hunch to go by, why would the university take you seriously and is this really a hill you desire to die on? You are putting a lot of professional reputation on the stake here while you don't even know if there is a problem.

@Allure There might also be the issue that you are abusing confidential information - is the journal even ok with you discussing details of an under-review manuscript with outsiders? I would definitely check this, and my assumption is that the answer will be "no".

That's why if I do anything I would presumably confirm if there is a problem first with the student. Sort of like this: https://edition.cnn.com/2021/01/15/us/orlando-waitress-boy-abuse-note-trnd/index.html (which would also be what option 3 would be about)

@Allure interesting case with the CNN news you linked. Although I'd say the situation is quite different in yours. For one, there are more and stronger evidences in the CNN news. Second, the waiter can insert their concern via a normal interaction with the table. If, for example, the waiter had waited until they left and only then felt that they should do something, there wasn't anything the waiter could do. Following them  discreetly into the to their home to observe whether there is indeed an abuse is not appropriate. I guess that's what xLeitix is trying to say with your Opt 3.

I find it very easy to imagine a scenario where the student did most of the writing even if the professor contributed most of the ideas.
Reasons for this include:

the professor is likely to be busy with other things;
the paper may well be low priority for the professor but high priority for the student;
there may be a university requirement that the student does almost all the writing in order to get (internal) credit.

These reasons are essentially independent of scientific contribution.
If you had any evidence that the contributions were being misrepresented, perhaps you should act on it. But you don't.

Please do the last option you have suggested. That is, to write to the student co-author directly and exclusively and find out the co-authors' respective contributions.
In Sri Lanka, I know there is a monetary reward scheme for academic staff called "Research Allowance" (since 2011), which is an additional monthly payment of 33% of the basic salary for one year for research undertaken during the course of a year (it need not be necessarily published). Many academics steal students' undergraduate dissertations (without their knowledge) to claim this Research Allowance. Thus, there is widespread abuse of this incentive scheme. In Sri Lanka, the bulk of the universities are state-owned/funded,

I agree fully with xLeitix’s answer, that your description of the situation doesn’t sound like such serious cause for concern, and certainly doesn’t justify taking unilateral action.
Here I’ll add one more point: at least in the systems I’m familiar with, any action should initially go through the submission’s handling editor.  They are the person who’s in contact with the authors on non-administrative questions, and who is primarily responsible for making sure the paper meets the journal’s policies and standards (including for authorship ordering) — and they should know the conventions of the field well enough to know those standards well.  So if they are concerned about this point (either on their own behalf, or because a referee has raised the issue), then it’s completely appropriate for them to raise it with the authors to figure out whether there’s any problem, or (if they’re convinced there is a problem) to take some action in response.
However, from what you write, it sounds like you’re not the handling editor.  If that’s the case, you have a clear path of action: mention your concerns to that editor, and leave any further action to them.  They, not you, have the responsibility and authority to handle it — and hopefully the experience, integrity, and good sense to handle it well.  And hopefully they’ll be able to either reassure you that the current ordering is appropriate, or else keep you updated on whatever further inquiry or actions they take.
(On the other hand, if you are the handling editor, then that’s important information you should add to the question!)

I want to suggest two things:
(1) - Do nothing. You have far too little evidence. I work on the boundary between maths and computing, and there we often put papers in alphabetical order of author. Also, maybe the professor just isn't an expert in this area, worked on something with a student, then wrote it up badly. Unless you have significantly more evidence, I really don't think any response here is approriate.
Extra: If you want information like this, ask for it! Your journal can ask for a written statement on what each author contributed to the paper, which can either be kept private, or included as part of the paper. Many journals require such things. It might be hard to retroactively apply that to this paper, but it would help with similar future issues -- while it doesn't fix outright lying, in my experience if greatly helps with such problems.

I don't think that you or the Journal should contact the corresponding author (who, I presume, is the first author listed) and bring this up directly.
I do think you might discuss this with the Editor or the managing editor involved with sending this out to reviewers and determining the publication status.
It might be that the student (who ostensibly wrote the paper) wanted his professor to be the principal author to give the paper more gravitas than it might be worth.
The paper should be reviewed and judged purely on its merits, including writing style and other pedantic issues.  An honest and critical review might embarrass the professor enough that he/she gets more deeply involved in the revision of the paper.

Given the current state of deception and fraud in scientific research, I would have a hard time ignoring it and choosing option 1. Doing nothing really just contributes to breaking the system further.
The letter outline by Mitsuko seems like a good option.

One more vote for Option 1 - "do nothing".
Besides all the other things mentioned in the other answers, there might have been some kind of agreement between the authors. During my time as a PhD student, I once worked on a paper with a maths professor, and (to me) it felt like our contributions were roughly the same.
When it came to actually submitting, the professor asked me, if I needed a "first author paper" in order to graduate (this was the case in the maths department). Since I didn't (there was no such requirement in the engineering department) and at the time was already planning to leave the university after my PhD, I happily gave up the spot as first author. To me, being first or second author made no difference, but for her, it meant having to write another paper or not that year, in order to fulfill her agreement with the university.

I'm sorry to say this, but if author ordering in your case exclusively relied on the described agreement between the authors and was totally decoupled from contribution to the paper (and that amount of contribution was indeed noticeably different), it sounds like you may have engaged in a scientifically unethical scheme together with your professor. This is not to say that I think the "agreement with the university" for your professor is a good thing (after all, it gives rise to exactly the described case where the metric is gamed), but it may not be desirable to use your case as a ...

... precedent for deciding on a course of action in the question at hand.

I am a recent graduate of a Japanese university, and while I may not be expected to respond to such a question, I feel compelled to express my thoughts because I myself was taken advantage of as an undegraduate student.
If it's evident that the student did most of the writing, it is very probable that they did most of the intellectual work and deserve to be in the first position. It's not very common for a professor to do most of the intellectual work and let their students write the text about the work done by the professor. Although one of the answers lists possible scenarios in which this could happen, I have never heard about this actually happening.
And I have heard about numerous instances of students being taken advantage of. I know one professor, albeit not in my country, who used to tell his students that a good traditional way to arrange authors is alphabetical. So, all or almost all papers written by his students had the alphabetical order - and the professor's surname starts with B. And I know a professor who used to tell his students and staff that in order to secure good funding for his group, he must have a lot of first-authored papers. He is the first author of a majority of papers published by his research group.
In your question, you make it clear that the writing style and the research conducted say it all. So, if the research looks like a student project, AND the manuscript looks like written by a student, this is a huge red flag. And you know this, because otherwise you would not have asked this question, in the first place.
It's very easy to play the Devil's advocate and say that nothing wrong might be taking place, but you know what the odds are.
So, please, don't be a silent accomplice to potential wrongdoing. You cannot imagine how grateful the student could be if you stand up for what's right.
I suggest writing to the authors with a message of the following kind and CCing the student:

Dear Prof. XXX and Mr. XXX,
Thank you for submitting your manuscript to the Journal of XXX.
Unfortunately, there's an issue that presents a challenge in accepting
your manuscript for publication in our journal. The concern pertains
to the author order on the author list. While I understand that you
may have valid reasons for the chosen arrangement, it is potentially
harmful to the reputation of the journal to accept papers that don't look like conforming to common conventions. This includes papers that are
first-authored by an experienced researcher, but look like
mostly written and contributed to by a student guided by that
researcher.
Therefore, if you have strong reasons for the chosen arrangement, I suggest adding a statement of the authors' individual contributions in order to dispel any concerns our readers may have. Alternatively, you could discuss the possibility of swapping your positions in the author list.
I sincerely apologize for any inconvenience caused and appreciate your
understanding in this matter.
Yours sincerely, XXX

I think that sending such a message is pretty safe because it doesn't say the student is being bullied or wronged. In fact, the message says that you understand the authors may have valid reasons for the chosen arrangement, and you offer them the option of explaining their individual contributions.
I hope that my answer will be at least helpful in understanding what the student might want you to do.

I upvoted, but are you sure doing this is a good idea? I don't actually know if the student is being bullied; it's suggestive but very far from conclusive. It feels like I ought to confirm that first before taking any drastic action.

@Allure I think that sending such a message is pretty safe because it doesn't say the student is being bullied. In fact, the message says that you understand the authors may have valid reasons for the chosen arrangement. The only argument the message contains is the concern about the reputation of the journal. I don't think that the professor will have any valid reason to complain upon receiving such a message. At most, he might try to persuade you.

@Allure If you message the student only, you will probably not get any insight. Students are usually heavily dependent on their advisors. The first thing the student will do in response to a private message from the editor is most likely talking with the advisor.

@Allure Also, many students do not even realize that something wrong is going on. I know one professor, albeit not in my country, who used to tell his students that a good way to arrange authors is alphabetical. So, all or almost all papers written by his students had the alphabetical order - and the professor's surname starts with B. And I know a professor who used to tell his students and staff that in order to secure good funding for his group, he must have a lot of first-authored papers.

Downvoted. Providing a fake reason for rejecting the paper is a terrible idea, as is approaching the situation with the attitude that your speculation is reality and the only options the authors have are to either admit that your speculation is correct or submit elsewhere. All around bad idea.

@Allure I think the key is to use the right wording: "papers APPARENTLY not conforming to common conventions," "despite SEEMINGLY being mostly written and contributed to by a student." And it's important to use a polite and professional tone. If you do all that, you are in the clear.

@Mitsuko This is not at all how actual communication works and, irrespective of the question at hand here, it makes me worried that you will get into some very unpleasant professional situations if you think you can just throw serious accusations around while imagining that you "are in the clear" because you used the words "apparently' and "seemingly". For example: "Mitsuko SEEMINGLY does not know how academic communication works, APPARENTLY as a consequence of her lack of experience." Did the two words really change the message? No, not really, it's still insulting and obnoxious.

@AdamPřenosil I've just edited my answer, taking into account your comments. I removed a paragraph about rejecting on the grounds of limited significance, and now the proposed message includes a suggestion to add a statement of the authors' individual contributions in order to dispel potential concerns by readers.

"If it's evident that the student did most of the writing, it is very probable that they did most of the intellectual work" - from my own experience with including undergraduate students in research, I have to say this really does not follow at least in applied CS. I have witnessed undergraduate projects that resulted in papers that almost always played out like this: The supervisor brings in the overall topic, has a concrete research question in mind, and outlines this context to students along with a possible solution to test. Students implement a software prototype and, together ...

... with the supervisor, conduct a small user study designed by the supervisor. The students are given the opportunity to participate in a paper, whose text they draft based on the supervisor's outline. Typically, undergraduates do not have a good "feeling" for which aspects are interesting to the scientific community, and may not even have developed a good grasp of the abstract problem. They thus try to "enhance" their text by going in (too) much detail about the more basic aspects, which they do fully understand. The supervisor then, despite having done most of the conceptual/scientific ...

... work, does not want to spend the time to rewrite the paper, also because that would reduce the students' contribution to merely writing source code and noting down answers in a user study, and just fixes/completes the most crucial parts. (And I've been on both sides of this.) Summarizing, I do agree with your remark about red flags, but red flags are, after all, no proof, and can easily be "red herrings" in this situation.


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What would one write in a paper describing a counterexample to a published proof?
Inspired by What steps should I take when contacting another researcher after finding possible errors in their work? where the OP describes something similar.
Let's say I've found a counterexample to Fermat's Last Theorem. The theorem is easily understood, and any counterexample can be quickly verified. It's also been proven as true by Andrew Wiles in 1994, so a counterexample would imply an error in Andrew Wiles' proof.
What should the paper I write giving the counterexample look like? Do I need to give a historical background of the theorem? Do I need to find the error in Andrew Wiles' proof?
I'm looking for an example where the counterexample is easily understood with high school mathematics (as in the linked question), and where the conjecture/theorem has a published proof. The closest paper I'm aware of  seems to have been abnormally short, and Euler's conjecture did not have a 'proof'. I vaguely remember reading about a proof that the cube root of some number is irrational that was proven wrong by a counterexample, I believe in one of Martin Gardner's columns for Scientific American, but this seems to imply that I remembered wrong. Other counterexample papers I found from a Google search all seem to be quite complex, and it takes real effort to prove that the counterexample is a counterexample. If there is no such example, a "best practice" description would also be helpful.
Footnote: I don't actually have a counterexample to Fermat's Last Theorem.

So you're looking for simple examples of counterexamples to conjectured theorems? The  case of the Euler characteristic comes to mind, but I suppose counterexamples published in 1812 aren't ideal for your purposes.

@Anyon fascinating link, thanks for sharing. It might not be directly related to this question, but it was illuminating.

A famous counterexample in knot theory is the observation by Ken Perko that two knots long thought to be distinct are in fact equivalent; On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262—266. The paper is mostly about classification of the ten-crossing knots according to their linking numbers, but along the way Perko notes:

This single result is certainly elementary, in that anyone could look at the sequence of pictures - possibly aided by a piece of string - and confirm that the manoueuvers work. Although it is not explicitly stated in the paper, the observation also disproves a "theorem" of C. N. Little in Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), who claimed (p774):

Theorem: The total twist of a reduced knot is constant for all forms in which the knot can be projected.

or in modern language, that the writhe is an invariant.
An answer on Math Overflow by Daniel Moskovich gives more detail.
Perko does not explore exactly why Little's proof went wrong, perhaps because it was from an early era of knot theory where much of the approach had already been superseded, or regarded as insufficiently rigorous in general.
An FLT counterexample would be at the other end of the spectrum since Wiles's proof has been closely scrutinized, and connects with many other mathematical results that are well-studied: so it would be more surprising to find out that it was wrong. In such a case it would be very interesting for mathematicians to understand exactly why the original proof went astray. But I expect that because of the theorem's prominence, a counterexample would still be publishable without such an account.
But in general, I'd expect a counterexample paper to say more than "here is the counterexample". For example, if it was found by computer search then there may be interesting things to say about how the search was done. There could also be results like "any counterexample must have such-and-such form", or "the original theorem is still true for quasiregular bloboids, but fails for the general case". Even if the eventual counterexample is easy to check by elementary means, these additional bits of context can account for some of the complexity you've noticed. (Similarly, the main results of Perko's paper involve words like "autohomeomorphism" even though the counterexample is in terms of "look at this picture".) This is fundamentally because a paper showing a counterexample is still a mathematical paper to be judged by the same standards as any other, regarding noteworthiness, motivation, rigour, etc. These factors also relate to where a paper might appear.


text:
How to most effectively get good at a new topic?
I have recently had a change of interest and would like to focus on a different area within my field. That being said, I want to stress that the area happens to differ quite a bit from my current area. I am not too far in my studies, so it's not like I am changing my PhD thesis midway or anything like that.
However, I would like to know how I can most effectively get to grips with my new topic. I am obviously not completely new to it, but I have so much more knowledge in my old topic and am now worried that it will be a disadvantage, because I might be competing with people for PhD spots that have had their focus for many years and thus much better writing samples etc.
Any advice on how to become good relatively fast?

I will assume that by "becoming good" you actually mean acquiring knowledge in the field?

If there were a special trick to "become good fast" then wouldn't everyone do it? Maybe they already are? Maybe "fast" is not as fast as we'd wish, and "good" is farther in the distance than we'd imagined?

Hippocrates already said it: "ars longa,vita brevis" (sorry, I haven't the original greek at hand). It translates into "to learn the art takes a long time, life to practice it is short"

@paulgarrett maybe everyone is, but I don't know about it? Also, I have not asked for the magic pill here that makes me jump the queue. But it is certainly true that some approaches to a new topic are more fruitful than others. (Although maybe not for everyone)

The simple answer is: Review Lectures, then read the referenced books / papers (in this order).
There are a variety of good lectures on free course websites out there. (For example on youtube or coursera.) It'll give you a good introduction and - more importantly - will tell you what is relevant to know about in this field.
If you can't find a lecture or at least slides, try to find a few (recent) PhD dissertations in the field and read their introductions / related work / background chapters. These are usually available for free and they will give you a good overview of what is considered relevant.
You could then move on to study recent research papers, but - depending on your prior knowledge - you may not know half the methods that have been used here.
In any case, find out which are the most relevant conferences in this field and see which Professors publish there a regularly. (The Prof. is usually last on the author list). From there you'll hopefully find the most key people in the field and check out their websites and latest publications to see what they are working on.
I'll have to stretch that the right approach will largely depend on how well you are familiar with the basics of the field already. You may have to study a few books or some additional mathematics course work relevant to this new field you are interested in.
Since you appear to be going to university - maybe it's also a good idea to just ask the Prof. most relevant to this field on campus for references and guidance. They will most likely know who's book and publications to read. Maybe this should actually be the first thing you want to do ...

Try this:

Read a lot of original papers in the field, ideally by the best and/or pioneering people in the field (they usually write better and they know why they are doing particular things, rather than regurgitating it from textbook knowledge).
Then go to a few of the best conference(s) in the field and listen to as much as you possibly can. Visit posters, talk to people, as much as you can handle.
If you can find a friendly local expert, buttonhole them to explain the big picture to you.
Then give a seminar or seminar series on the topic.
Write mini "lecture notes" for the seminar you gave. Estimate 8 pages for 2 hours of seminar.
Voila, instant-expert! (well, not quite, but you now have a head-start).

#2 is a good idea, but I assume it's difficult for many master students to afford a trip and pay high fees often associated with attendance of the top conferences. Yes, they usually have student prices, but you have to be really dedicated - I also doubt it will be the best approach, since those meetings are often more networking than educating people. They also generally don't discuss the basics of the field there, but if you have the means to do it - go for it :)

I talk from experience, this is what I did. On #2, you are right: it is expensive. I was advised by a sponsor to go to my first conference ever, and was promised significant travel support for a report in return, so I dared to go (and did as stated above). I came back, and reported, but the sponsor conveniently "forgot" his promise even after multiple reminders, which ripped a big hole in my finances. Nonetheless, I never really resented it, because the advice was worth the hole in my pocket. I used conferences for networking later, but my first ones were predominantly for learning.

Sorry to disappoint, but the "toil away at a topic for 3 to 5 years until your thesis is done" is the fast track.

I did not ask for a fast track. I am just interested in ways to organise yourself which might be better than others when e.g. approaching a new topic. See also my comment to paulgarret

@KenGoldhaus well you did use the words "quickly" and "relatively fast" in your question. By your clarifying comment about not asking for a fast track, it now sounds like what you're really asking is "how can I get good at a new subject at a normal rate?" which, as other have pointed out, is a very standard question with standard answers, so you can hardly expect people to offer some amazing new insights here.

@DanRomik I agree that I should have maybe not used the phrases 'quickly' or 'relatively fast'. Let's say 'effectively' then. Speed is often an indicator of effectiveness, so that's why I wrote what I wrote. No reason though to assume I am talking about a fast track. I hope it's clearer now. (also, let's not forget the word 'relative' there. but enough)

I disagree. Part of the objectives of higher education is being able to familiarise oneself with a new, related topic, in a reasonable time frame, by drawing from what you already know.

@KenGoldhaus, my comment stands. The best known way to learn a new topic effectively and in depth is as I state.

@Davidmh, part of learning effectively is certainly training in how to separate the chaff from the wheat, and to assimilate new things.


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Prerequisites for Master's Degree in Pure Mathematics (and eventually PhD)
This will be my first post on the Academia SE so apologies if there are some conventions I break!
I am an undergraduate student of mathematics at a university in the UK whose emphasis lies a lot more in statistics and applied mathematics than it does in pure mathematics, there being only 2 modules in the entire degree program that focus only partially on higher algebra, and only 4 weeks of the entire 3 year program that have anything to do with number theory. I am in this position because my A-Levels did not go as well as I had expected as I had a few issues and did not achieve the grades needed to go to a more "pure based" university. I am deeply interested in the area of Algebraic Number Theory and intend to go on to study a master's degree in pure mathematics and eventually/hopefully a PhD.
My question is this; what should I/can I do to "boost" my application to a master's degree in pure mathematics if my undergraduate degree has a relatively low pure mathematics content?
I will be applying for an undergraduate research bursary from the London Mathematical Society to fund an 8 week research project, be it expository or otherwise, in the field of Algebraic Number Theory.
Thank you in advance for any information you can give me.

@user170039 I did! I got admitted on a scholarship to the University of Heidelberg in Germany. I essentially self-taught all of the relevant stuff for my undergrad dissertation and got a high enough grade that I was accepted for a scholarship etc. :)

Although I am doing a PhD in the field of Computer Science, I of course cannot give specific advise on what you should do about mathematics MSc, but here are some of my observations on the general case.
In the last two years, I have applied a lot of PhD programmes across Europe. I have been accepted to one recently. And that is probably one out of fifty if not more.
As far as I have observed, the priority is not a student with "a lot of knowledge", but a student "who can study hard on the given task." This observation is based on the profiles of accepted students instead of me (stalker alert).
Some of them have lower GPA, and almost none of them have publications. I think what they somehow proved that they can study harder and be more productive.
The interesting part that matches with your question is, both their theses and their stated area of interests were less related to the research area.
Surprisingly, someone who took his degree from mathematics was accepted to a research group that focuses primarily on application rather than theory.
If we switch from PhD to master's degree, I don't really believe having a relative background should have more weight than being a hard-worker (of course I am not talking about another discipline like psychology) in the process of decision.
If I were you, I would direct the exactly same question to a professor from the department you want to apply. Questions like

How can I convince you that I am good enough to pursue a master's
  degree?

or

Are there any specific criteria you are seeking such as having a
  relatively strong background on pure mathematics?

would do no harm. I think being deeply interested as a master applicant in a research topic puts you one or two steps further. Of course you somehow have to show that you are actually deeply interested (this is the hard part that I could not figure out how).
In short, do not over-try to "boost" your application and do not hesitate to contact one of the faculty members.

Thanks a lot for your answer. I contacted the professor directly to ask him what I should do and he said that the pure maths content of my degree should be okay in principle. He said that the fact that I am self-teaching large areas relevant to my interests would support my application and that I shouldn't worry too much about it. Thanks again!

@Ed_4434 I am very happy that my advise worked I could help! Hope that you get the acceptance that you want.

For Algebraic Number Theory you can try reading up the topics of group theory, ring theory, field theory, Galois theory. Perhaps write up some notes using LaTeX and solve some of the exercises in books. Showing the notes/exercises to the professors should be quite impressive.

Thanks for your comment, I've already been self-teaching myself some basic ring and field theory with the aim of understanding Galois Theory and upwards! :)

All the best and good luck!

authors: ธัชชัย มีเดช


text:
How can I make my thesis supervisor be more responsive?
This question is related to this one and this one. My Masters' thesis supervisor is incredibly superficial as in he doesn't respond to messages in a timely manner; I am not talking about understandable delays such as a few days or perhaps even a week but 15 weeks all in all.
What I have tried up to now is to have a professional discussion about my progress so far, feedback and future directions. All seemed well, I received feedback regularly (for two weeks) and then I had to redo the discussion. This wouldn't normally count as an issue but he is incredibly inconsistent in his feedback. For example, the suggestions I have received this week are inconsistent with what he requested the week before, but consistent with what was needed two weeks back. There is always this back and forth between what I should modify within my thesis (he doesn't keep any records of what he suggested and always ends up in giving advice that contradicts what he mentioned previously). I've resorted to taking pictures of the sheets (sometimes feedback contains figures and tables) and actually including them in my thesis draft prior to sending it to him. Another issue is related to his organization of these drafts. Somehow, he never seems to read the latest submitted version the thesis draft, doesn't sort files by date, etc.. As a solution, I am now prefixing the thesis name (the name of the .pdf file) with the date in which I submit it. Unfortunately, it didn't help (I'm still at square one).
Sometimes feedback is lackluster, he regularly takes a week to read two pages and give advice on them. The problem is that they are from a chapter that has long passed review (and was agreed upon by him that it is correct, both theoretically and grammatically).
Any help as to how I may actually complete this thesis? It's pretty infuriating because I've finished with moderately high grades and am now stuck because of this supervisor at the end.

Can you estimate what the problem is? Is he lazy, overworked, burnt-out, uninterested, or just plain disorganised? This informs your action.

Mostly lazy and disorganized.

You could put in a section: "You requested, I did:" where you list what he asked you to do (bullet points only, short sentences, easy to read). This is a classic example of "managing your manager", and here it is really needed. Can you do Dropbox? This way, you could make sure that the directory is always organised in your way. "they are from a chapter that has long passed review" - why does he have to read that again, then?

I have already done that. He does not read them :(. At this rate there doesn't seem to be a definite ending date for my work.

Can your university allocate you a second supervisor?

Sadly, no. I have already asked about that and it would seem that I'd have to rewrite my thesis from scratch if I would be allowed to change the supervisor. He is technically not a professor but an employee at a company that is indirectly affiliated to the the university

Well, they are responsible for your wellbeing. What do they suggest? They cannot leave you hanging around like that...

They mentioned that I should keep in touch with him. I am sending him emails every two days in a hope to get an answer. I don't think it can hurt at this point.

Could it be that you are soliciting micromanagement? Your supervisor gives advice, until you give him the final version, which he then grades. Discuss his advice with him, defend your own solution if you find it better, or simply ignore it if you think he's wrong and will understand better next time, after you've changed some small point.

@Karl I've submitted my thesis. It was incredibly difficult working with him and am no longer pursuing a PhD; I've decided not to invest any more time in academical ventures since they are high risk/low (or better yet no) gain (I've had enough of bureaucratical mazes and incompetent people that shouldn't have gotten into a teaching position to begin with).

First of all, any relationship is a two way street. You can not blame everything on your supervisor, you are the one who chose him/her; and you are the one that should have request regular meeting with him/her in the past. You have two options:

Managing the Situation: You can start managing the situation and start requesting regular meeting with him/her. He/she is not responsive on email? Well, you can see him/her on his/her weekly office hours. Doesn't have office hours? Well you have to pop in into his/her office and see if he/she is there and kindly ask him/her to respond to your email. You see what I'm trying to say here?
Find Another Supervisor: You did take classes, and you saw the lecturers. You can always find a supervisor that does have the "common sense" part and kindly ask/her to do the master thesis with him/her instead. There are always young and talented lecturers (and not yet popular) that do take such cases at least because to gain some experience and also help such students.

Note: At the end of the day, you need to write your own thesis, and you are the one that present/defend your thesis at the end. So also pay attention to your own work and try to finish it ASAP as any supervisor has some limitation to help you figure out your own path.

Well, in my case no. I was given an assignment, a deadline (which is now extended by a few months) and a supervisor. I didn't have any freedom of choice in that regard. He is not a lecturer but an employee of an external institute to the university. I am writing a thesis in that company, not in the university or working with people directly affiliated with it. Changing the supervisor was and is not possible. My issue with working on my own is that he always wants me to change what I've written. From an 80 paper work I'm down to 58. It is very difficult for me to get what he wants.

@Sebi Well as you are saying this is your only option. Then, you need to manage the situation by having regular meetings with him, show him what you have done, what you are willing to do, and why his approach is not fitting because you have only couple of months. At the end of the day he is your supervisor/boss and you need to maintain the situation and his comments.

First, you need to accept that you can't make them anything (nor anyone else for that matter).
With that out of the way, let's discuss what you can do in the situation. Your adviser is obviously either way too busy or just too easy-going. You made some good attempts to remedy the thing. (I'm not going to suggest to find another supervisor, because you won't be always in a situation where you can afford to do so and you'll rather often encounter such people you need to collaborate with)
I had a somewhat similar situation regarding the conflicting requirements a while back. After the meeting, I would just compose an email with the bottom line(s) of the meet-up. This email would contain a list of action items that need to be addressed before the next meeting, with the person responsible for them:
subject: action items

read XYZ paper, see how it relates with ABC experiment (you)
extend x.y table with something (you)
check with prof. someone to schedule you a slot on the lab equipment (your adviser)

This needs to be short and concise. Before long, you will have a steady email stream of a written trace of your requirements and their progress. It is in one place, chronologically ordered. Feel free to include your thesis versions along.

I have already done this. He doesn't seem to take them into account.

the point here is that he doesn't really need to, it is you who takes care of them. when you go to a meeting, the first subject should be "last time we discussed the following things...". This approach should work, if the adviser needs a kick start with pre-chewed information, because he is overburdened with other things (disorganized, overly busy, etc.), but still does care. It is important that he needs to do nothing for this to work, no replies, to emails of his own, he doesn't even need to read them. You just need to bring the last mail to the next meeting.


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How to handle unprofessionalism in academia?
This question is in part related to this question and this one. In my case, I’m part of a double-degree Master’s programme¹, and am having a lot of difficulties at my German exit university. There are two coordinators/supervisors:

one that handles more of the organizational aspects of studies (recommends courses, seminars, internships, thesis topics, etc.);
one that takes care of all matters concerning thesis work.

Both have been incredibly unprofessional; because of the first, I was the only student who:

was not paid for his contributions while working on the thesis, the others enrolled at the same university and company (we have to develop our thesis project in an “outside” company)
due to some bogus reason, which was not made entirely clear, had to retake a seminar while working on the thesis even though it has been passed. The main issue was that I had to commute from the company to the university in order to participate in it.

Other aspects include:

I have to deal with inconveniences such as being called to the coordinator’s office for meetings scheduled by him, that he would not show up at (this can become incredibly infuriating after a while). When contacted, he mentioned that he was on a trip for two weeks.
Extreme favoritism towards other students. There were students who did not finish their thesis, but “graduated” nevertheless.
Very, very high unresponsiveness to e-mail messages (I always keep them short and to the point – usually between 5 and 10 sentences).
Toxic attitude; when confronted with the situation, the coordinator mentioned that he doesn’t care. I stated that, because of him and his thesis recommendation I had lost little over 23000$ (the amount that the other students earned during the more than one year period of study/thesis work at the company) and a few months of my time. He again mentioned that he doesn’t care.

What would be the most preferable course of action to take?

¹ The first year of my Master’s studies is spent at one university while the second at another. The second university is the one at which I’m working on my thesis. The double degree means that I will get a degree from both universities, when I graduate at my current university.

How long was the seminar? Hours? Days? Weeks? Months?

2 hours a week for 6 months. Also, it required one to go through a number of papers each week and write one at the end.

Which country was that in? If you pay for your degree, you can be expected to receive a reasonable service and a reputable university should be able to provide recourse mechanisms. Unless, of course, you happen to be at "Dump University" (add an "r" at a suitable location).

Germany. I could not find any suitable recourse mechanism. Most require involvement in local laws which I'm very unfamiliar with (and I cannot afford a lawyer).

Could you clarify what the "double degree programme" means? Germans generally seem to keep the bachelor's, master's, and PhD as all quite separate steps (unlike, say, US, where MSc+PhD is one long 5-year haul).

Yes, in my double degree programme, the first year of my Masters' studies was spent at one university while the second at another. The second university is the one at which I'm working on my thesis. The double degree means that I will get a degree from both universities when I graduate at my current university in Germany.

Ah, Germany. Well, I am almost tempted to say, ask for the money back that you paid to the university, i.e. probably none, unless you are in a private one. I do not see many options for you under these circumstances; German universities try to keep a very high degree of autonomy for the profs which, if you are in a good group, is fantastic (anglo-saxon universities have lots of red tape stifling autonomy), unfortunately, there is no lower bound limiting how bad it can get, either. You have the bad luck being in a bad group. Keep your head down and get through it ASAP.

In my limited experience this is the BS many have to deal with in academia, which is the main reason why doing a PhD made me change my mind about working in academia.
Universities are a place where bureaucracy thrives, however this can be to your advantage, when it comes to complaints procedures, which are taken very seriously (usually).
Find out the university's official complaint procedure (usually two stages, first stage is dealt by the school and if issues continue you may raise it to university level, at which point the supervisor, if at fault, will be dealt with in a biblical format ;).)
Make an official complaint, provide all of the evidence required (including emails unanswered), and it should be dealt with withing a couple of months.

This is a reasonable answer, but -1 for "it should be dealt with within a couple of months", which sounds like an unrealistic prediction that would set OP's expectations up for a possible disappointment.

Sorry, this is the timeline published on my university's complaint forms only, you are correct.

Without further details, I see essentially three courses of action:

Talk to your student union (Fachschaft). They are much more familiar with your circumstances and should know the organisational structures and know where written rules were broken. They can escalate things for you (and keep you anonymous as far as possible) or at least give you better advice than we can.

Escalate your situation to someone superior to your supervisor or coordinator – which would probably either be a superior program coordinator, a department head (Fachgruppenvorsitzender) or dean (Dekan).

extreme favoritism towards other students. There were students who did not finish their thesis, but "graduated" nevertheless

If this part is true, it can likely be blown up on its own. If somebody graduated and no proper thesis can be found, this is very bad for whoever graduated them. An investigation of such cases alone might lead to your supervisor and coordinator being much more careful about what they do.

In all cases, be aware of the possibility that taking action might lead to repercussions. In particular, the superiors of those people may be equally unconcerned about these issues. The student union should be more safe and the first you should contact anyway.