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Why would a school prevent reapplying after declining an offer of admission?
I am currently applying for a mid-range private school that teaches the career I want primarily (software engineering) and a prestigious public school for computer science (it doesn't have software engineering). So if I make it into the public university, I would evaluate between the specific career of the private one and the economic help of the public one, and choose the most appropriate one.
However, reading their entrance rules, I saw that if I decline the offer of admission in the public one, I could not apply to it again for the next two admission cycles (at two per year, that means a year and a half). What is the philosophy behind this? Applying to several schools benefits  students, giving them more schools to choose from. And it also raises the funding because of the payment that is done to give the entrance exam (which is not one of the things public schools in my country can brag about). And it doesn't harm anyone, because the next one in the list would enter. The only collateral damage I can think of is extra administrative effort, but a year and a half vetted from applying to the university because of that seems too much for me. So, why? Is there some extra side effect that they are accounting for?

The information @Pete L. Clark requested: I am living in Peru, and here the admission process in public universities is pretty simple: You just present your legal and education papers, pay for the entrace exam, take it, and see if you are admitted or not. So, everything boils down to that 3-days exam. But once it is finished, you are in.

@aeismail I just changed a couple of things that I didn't mean, but thank you very much for the edit, I even learned from it!

I am curious about exactly how this restriction is worded. That might present a clue.

We already have one answer explaining the perspective of the US academic community.  But in the comments to this answer we learn that the OP is applying to universities in a different country for which the system is different (e.g. is there a pre-admission exam given by the university? That does not happen in the US).   If so, please modify the question to provide more specific geographic information.  How can people answer it usefully if they don't know what context you are talking about?

@JR It is just given as-is:The applicants who decline their vacant cannot participate in the next two admissions(But in spanish).

@PeteL.Clark I agree, I added some information(mostly the same as the comments)

I believe the issue here is that if a student has already been offered admission and turned it down, then the school has already given that student due consideration in reviewing the application. Reapplying in the following year means that the candidate was apparently unsatisfied with any of the offers of admission received, which included the school in question. So what would have changed in such a short period of time that meant the offer wasn't good enough then, but is good enough now?

Well, one of the things that would make me reconsider their offer is that I couldn't sustain ong to wait for the next exam. (This is not a problem really, because I am planning on staying their offer if I make it. I am just curious about such a condition...)

Programs don't like to be used as a "backup", because offers to people using them in this way delay offers to others, sometimes to the point that the other people give up and accept offers from yet-other places. Thus, making offers that fail tends to degrade the quality of candidates who will accept offers. Thus, since you declined once, obviously chances are good that you'd decline again, thus again somewhat-lowering the quality of candidates that will be made an offer early enough to accept.

Actually, the entrance to public universities is simpler in my country. You just give your studies certificates, pay for the exam, give the exam, and see if you are admitted or not. But I agree that it could cause some problems for the ones that weren't given an offer at first because of the ones who declined it.

Ah, yes, my comments were specific to my experience in the U.S. (30+ years). But possibly people think in some such terms even if it is not so visibly part of the mechanism as in the U.S. Not to mention the "once burned, twice shy" riff, or the "offense" at being refused a "date" upon offer... for kids. This is human nature, I suppose, and some image of it will appear in otherwise-high-tone [sic] situations.

@paulgarrett I tend to think the human nature thing is very unlikely. Japan has an entrance system similar to what chubakueno describes. But I have never heard of forbidding reapplying or anything of that nature in Japan. If a university does such a thing, it would be perceived as being egregiously unfair. There are various valid reasons that may make applicants change their minds, e.g., serious illness (and unexpectedly quick recovery), a sudden change of their financial situations, and a whole lot of other personal issues. (cont'd)

I just googled in Japanese what my countrymen think if a Japanese school treats differently applicants who turned down offers. Opinions I found through casual googling were nearly unanimously "There's no way a reputable school would do such a thing." So, if OP's country also respects equal opportunity in education, the reason may be more professional than just "Reason? 'cause you rejected me once. It's human nature." I think it's more likely that the system there (or the school s/he is applying to) has some mechanism that would degrade the quality of candidates if reapplications are allowed.

@YuichiroFujiwara: how common is it in Japan to apply for several universities? If the ethics (or application mechanism) is "balanced" in the way that there is not an egregious amount of backup applications so a decline and re-application by the student usually means that something like the reasonable causes you mention are the normal case - then of course there is no reason for restricting the reapplications. Things are different if there is a problem of about everyone sending multiple backup applications.

@cbeleites It's a difficult question because I need to explain a lot of things. But roughly speaking, when it comes to public schools (which OP asked about), you can pick only one school. And if you didn't pass their entrance exam, you can apply for one more public school. And that's it. The reason for no mass application is complicated. But basically, if you apply for a prestigious public school (which is the case for OP), the outcomes of your applications to private schools should come first. You only take the exam for a public school if your first choice is that school at this point.

I guess whether backup applications (and re-applications next time) are or are not a problem depends very much on how many prospective students do how many backup applications.
No problem if a small percentage of prospective students apply for one other university.
But if lots of the people who want to study apply at several places, you end up in a situation where the administrative part of the whole process runs into chaos because one or two more rounds acceptance letters and waiting whether the student accepts are needed. And of course, the chances that a student who ended up on the waiting list with you accepted somewhere else or meanwhile took a job because they thought it unlikely to get a place to study would be high. Thus, administration has a lot of additional work and also stress. You cannot start the application process too early (e.g. the final exam of the schools is needed). But you need to allow a reasonable amount of time for the student to accept or decline the offer of the univerity and students will tend to send the refusal late (or forget to send it at all) because they wait for acceptance of another university. If you are too late sending out the acceptance (for those on the waiting list), this is anywhere between stressful and impossible for the students to accept: they have to find housing and move, and if courses already started they also have to catch up with the learning.
Also, universities cannot overbook like airlines.
In other words, the situation becomes highly stressful for everyone involved:

the university has a high unnecessary workload (n "backup exams" have to be graded for each student),
university administration also has a high workload and pressure: they need to react immediately on the refusal letters of the students and may need several rounds of acceptance.
Students who do not get accepted immedately at the university of their dreams have to stay ready for moving immeditaly without knowing where. Also the prospective student may "hang in the air" because they don't apply for a "regular" job / training while there is still a substantial chance of getting a late acceptance.

I'm wondering whether the fact that you have to pay for the exam is just another symptom of this same problem (or the univiersities' fear of that): both rules may be designed to keep down the number of backup applications.
There are other ways to try avoiding this administrative mess: In my country (Germany) the most overrun fields of study have a central application: you file your one appliation including a statement of backup universities (and backup subjects) and then get an offer according to this. If you end up at a university that was not your first choice, you can anyways start there and try to change later on.
Nevertheless, the last letters of acceptance are usually sent out when the semester has already started (to fill in places of students who accepted but didn't show up or because this is round 3 of the acceptance: the student on the waiting list declined).

For Erasmus Mundus master courses, anyone applying to more than two is automatically rejected for all.

@gerrit: but they surely have a clear statement of that rule!?

@cebeleites Yes, this is stated clearly.


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Listing Latin Honor GPA while you are still undergraduate in your CV
I am wondering if my GPA is in the category of one of Latin honors like Cum Laude or Summa Cum Laude but I am still undergrad can I include Latin honors in CV while applying to graduate school?
This post is talking about this case however whoever asked the question did not specify whether he is still undergrad or graduate student.
Honor Listed in undergrad. Should it go on the CV for grad applications ?

I would not include such a statement in your CV, since Latin honors are normally awarded at graduation. Until you have actually received the honors, don't list it in your CV. (Moreover, putting "eligible for magna cum laude" or the equivalent makes it look like you're trying too hard to list extra prizes, which reflects poorly on you.)

I think adding something like eligible would be fine. Undergraduate CVs are thin and I wouldn't look negatively on this.

I would.  I can read the number just fine, thanks.

@JeffE: Though the mapping "number -> latin phrase" is not well-defined, so "eligible for magna cum laude" gives you the extra information that this particular school considers this GPA sufficient for this honor.  Of course, you may or may not consider that information to be interesting.

So to conclude: based on what JeffE and aeismail and other said. should I include it in my CV or not?

@user59419 Clearly, they recommend you don't.

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Handling unsolicited proofs of famous mathematical problems
I have been receiving mails from (most probably amateurs), who claims to have proved famous mathematical problems, like the ABC Conjecture or Goldbach Conjecture. But invariably, they all contained mistakes. I decided not to waste my time on such unsolicited documents. But recently something interesting happened.
About 14 days earlier, I have received a mail from an Indian undergraduate student who claimed to have proved the Sylvester-Gallai Theorem in an elementary way. What is more amusing is that he claimed to have proved it using Mathematical Induction and a basic Euclidean Axiom. I decided to ignore it as usual. But yesterday I got his mail, telling me that-

I suppose you haven't considered my document worthy of your time and so you haven't gone through it at all, or it may be that you are so busy that you haven't found time to check your email account. If that's the case then just ignore this mail. But if it's the first case then I would like to tell you something.
Perhaps you have heard about the Indian Mathematician Srinivasa Ramanujan. He also sent his mathematical works to renowned mathematicians like Baker and Hobson but they didn't reply. Later he sent his manuscript to Hardy and his genius was recognized. But just suppose that Hardy also considered his work to be the work of a crank, without even going through it. Consider this be the case even if he would sent it to other mathematicians. How long could he continue sending his unsolicited formulas and theorems (which were without proof!) to other mathematicians and be rejected? Of course, finitely many times. After that, he perhaps wouldn't write to any mathematician even if he had, suppose for example proved the Riemann Hypothesis. Why would he? He was likely to be rejected.
So I suggest you at least to go through my document thoroughly and tell me precisely about it.
Please don't behave like Baker or Hobson.

What should I do now? Should I remain silent or go through the document? Any suggestion will be welcomed.

I have no idea how many such emails a mathematics professor receive every day, but I think you can give some of it as exercises to some undergraduate students who have an interest for research to find out the mistakes, it may be fun and a learning opportunity for them.

@StephanKolassa: Mods can't purge edit histories. You have to contact SE staff to do that.

I would suggest this gentleman to read up on opportunity costs and expected value.

Baker and Hobson are almost invariably omitted, or not named, in the canonical Ramanujan-Hardy story. The fact your correspondent mentions them makes me think his manuscript is worth a look.

I like the idea of farming it out as an extra-credit problem... and I agree that the badgering second note is a better reason to ignore it than to follow up on it. That's a marketing argument, not math.

I am reminded of the famous quote of Carl Sagan: "But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton, they laughed at the Wright Brothers. But they also laughed at Bozo the Clown."

@TheMathemagician: How does the correspondent's demonstrated above-average knowledge of a historical anecdote make it more likely that his mathematical manuscript is worth reading?

If I recall correctly, India's mathematics program was weak to non-existent back in Ramanujan's time. Now with Harish-Chandra Research Institute and Tata, there are plenty of people in India that can help him and give him feedback if it is not crank work. I would put effort in helping letters from people from less mathematically developed countries.

@Mark Meckes Well it doesn't, but at least you'll be dealing with a better class of crank. Seriously though, all the correspondent needs to do is post his two-page proof to mathoverflow.net and he'll get it critiqued for free.

@TheMathemagician: If by "critiqued" you mean "deleted within minutes" I agree with you.  Like most mathematicians, the MO community takes a dim view of anything that smacks of crankery, which this certainly does.  They also frown on "please check this proof" posts, preferring specific questions.

I had no idea how common this sort of behaviour was. I concede I am probably heavily over-estimating the probability of the work having any merit.

@TheMathemagician: It is extremely common. If you're interested in learning more about the taxonomy of people who submit crazy stuff to professional mathematicians, Underwood Dudley's book "Mathematical Cranks" is an enjoyable read.

Your description seems to hint -- but not say explicitly -- that the proof you got is without value.  The Sylvester-Gallai Theorem didn't ring a bell with me, so I looked it up on wikipedia and edited in the wikipedia link.  That article gives two different proofs, each taking up about half a page.  The first one (from a 1986 paper of Kelly) is completely elementary and uses a little Euclidean geometry.  Two pages thus sounds long for an elementary proof, but not necessarily prohibitively long.  Is it supposed to be clear that the student's work is of no value?  Do you feel this way?

@PeteL.Clark: Good thought to look it up.  I also assumed from the context that Sylvester-Gallai was a hard theorem.  If it is actually elementary, then it's more plausible that the student's work is correct, but it also makes the comparison with Ramanujan more absurd.

@Nate: I agree.  I also agree with the other answers that remark that India in 2014 is a very different place from India in 1912, and that if the correspondent is an undergraduate student that means he has faculty members who are being paid to (in part) evaluate his work and answer his questions.  I would myself feel more sympathy to someone who is nowhere near any academic environment and still trying to do mathematical research.

Inspired by some of the comments and answers here: If this is really such a nuisance, why not set up a platform somewhere between a Stack Exchange and classical peer review for this, where everybody can submit such proofs and the same people are somewhat encouraged (or forced) to review other submitter’s proofs? Also, everybody is free to review proofs. If any proof is any good, this should be recognised almost immediately. Everybody bothered by such requests can just reply with a link to that platform.

@Wrzlprmft: "If any proof is any good, this should be recognised almost immediately." Please teach me this skill of almost immediately recognizing whether proofs are any good! For lack of it, I must spend hours (or days...) going over every proof I see, including my own, those of my students and of my most trusted collaborators.  In the above cases, to a greater or lesser extent I have the advantage of being familiar with the author's writing style and being able to count on some shared knowledge and assumptions.  Reading a proof from a stranger who has not written many papers is much harder.

@PeteL.Clark: You misunderstood me. By “any good”, I meant that it actually requires the amount of time you describe to find a mistake or rule them out (which seems not to be the case for the submissions we are talking about here).

One strategy is to refer them to I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? and particularly this answer to help them phrase their emails better and understand your constraints and perspectives.

Not a total "on-topic" comment, but it may be helpful to somebody: I heard that an easy way to handle these emails is to answer like: "Dear [...], the manuscript you sent is not in my area of expertise [or insert some other excuse], but I know an expert on this topic. Please send you paper to [enter email of the sender of the previous unsolicited email you got] and he/she will be happy to discuss this with you." I heard that this worked in times of paper-mail…

Dirk, I assume you'd them give them the name of.your best frenemy in the discipline? :-)

Reading through all the responses, I'm glad that I'm in the social sciences. Our cranks are easier to figure out what with the references to Noah's Ark and the liberal use of ALL-CAPS.

I still think the "Museum of Unworkable Devices" website is one of the most elegant responses to this sort of thing I've seen. It specifically addresses perpetual motion machines, dividing them into related families and showing why they don't work. It may not stop any genuine cranks, but it's a good place to point those who aren't invoking non-physics.

"Dear [name]:
Your last email suggests that you do not understand how research results are disseminated. If you have an interesting result in your hands, you should, in this order, (i) discuss it with some local experts; (ii) upload it to the arxiv; (iii) submit an abstract to a conference; (iv) submit a paper to a journal. It is hardly beneficial to your research and your reputation to send it to a random person halfway around the world and then call them irrationally obtuse when they decline to read unsolicited mail from strangers from halfway around the world.
Sincerely, [your name]"

The Sylvester-Gallai theorem is in fact a canonical example of a problem that is easily solved using elementary geometry via the extremal principle, and the extremal principle is a basic consequence of induction. However, the clearest indicator that something is wrong is that the student can name the theorem (which implies he likely saw it on wikipedia) but cannot grasp that the extremal principle is essentially equivalent to induction. The other indicator is the arrogant attitude, which is typical of [semi-]cranks.

Unfortunately, I think there's little or nothing you can realistically do for most amateurs sending unsolicited manuscripts.  What they don't seem to realize is how common this is and what a bad state most of the manuscripts are in:

I average several amateur e-mails per week (and I shudder to think of how many Andrew Wiles or Terry Tao must get).  If I carefully read each paper and sent comments, that alone would occupy a substantial fraction of my professional activities, so I have to prioritize.
I at least flip through the papers, and most of them are obviously crackpot work.  Occasionally I see one that doesn't look ridiculous, and I try to be encouraging when appropriate, but I have yet to receive a publishable paper from an amateur.  The best I can do is generally to offer encouraging advice, and even that's uncommon.
Some people seem beyond hope (for example, the ones who send word salad), but some could presumably become solid researchers given the right education and mentoring.  However, this is not something I have a lot of time to provide.  I've got plenty of in-person students, some of whom would probably like more interaction, and I wouldn't feel comfortable telling them "Sorry, I'm busy trying to explain to some guy on the internet why his fuzzy understanding of quantum mechanics doesn't actually yield a short proof of Fermat's Last Theorem."  Even if the amateur seems promising, they aren't likely to be dramatically more promising than my students, and mentoring over the internet is less effective, so it's still an awkward trade-off.
Some amateurs react very poorly to feedback.  If you suggest their results are known (while complimenting them on their rediscovery), they angrily suggest that you must not have understood what they meant or are trying to deny them credit for their work.  If you don't believe their results, they accuse you of incompetence or laziness.  If you encourage them to apply to graduate school, they scoff at what academia would have to teach them.  This is of course only a minority of amateurs, but it's just common enough to discourage giving honest feedback: there's too much of a risk of feeling like you wasted time offering feedback to someone who only wanted validation and responded with insults.
Part of the problem is grandiose visions.  When people spend too much time daydreaming about being the next Ramanujan or finding the proof that didn't fit in Fermat's margin, it's really unsatisfying to learn that their story isn't actually as remarkable as they hoped.  It's much easier psychologically to move to the parallel story of the genius oppressed by academia, rather than starting an academic career from scratch.  (And even people who show no sign of grandiosity in their original e-mail sometimes have it hiding below the surface: I imagine that anyone who sends unsolicited accounts of their discoveries to experts is hoping for some degree of acclaim.)

So what to do about this?  In an ideal world, I'd give lots of time and attention to everyone who wrote, but these are scarce resources.  In practice, I handle it this way:

If the paper genuinely engages with my work and shows no signs of craziness (e.g., drawing religious conclusions from mathematics), I give at least a brief reply.  Same thing if I have some other good reason to believe it was sent specifically to me, and not just as one of many recipients.
If the paper looks relatively promising but has nothing specific to do with me, I'll reply if I have time and feel that the reply would be well received.
If the paper is on a topic I particularly know and care about but doesn't involve my work and doesn't seem especially promising, I might reply.
Otherwise, I probably won't reply, and almost certainly not if the paper deals with famous unsolved problems.

Continue to treat it as spam, and ignore it.
For every Ramanujan, there are many many thousands of time-wasters.
The reward:cost ratio, weighted by the ratio of misunderstood geniuses to time-wasters, is very very low.
If someone has any ability, they should be able to demonstrate it quickly. And if they've got any sense, they'll realise they need to demonstrate it up front to get taken seriously.
So if someone hasn't put a pre-print up somewhere (much easier to do now than in Ramanujan's day), and has no pre-published material, ignoring them is now an even safer bet than it ever was before.
In this particular case, your correspondant may have already tried posting on Math Overflow, although it may be someone else with the same name. Either way, if you're feeling generous with your time, you could prepare a canned response which went to all such neglected geniuses / timewasters that pointed them to Math Overflow, as a good place to engage with the Maths Research community and demonstrate that they are actually able.

I don't think he is the same person of whom I have been talking about. This person is an undergraduate. But it seems that the person you have pointed out is a professional mathematician. However, the comment- "..The reward:cost ratio, weighted by the ratio of misunderstood geniuses to time wasters, is very very low." is something useful to me. But do you actually have a statistics or it is just your intuition?

Just to play devil's advocate: how would you say one should demonstrate their ability up front, other than by sending a manuscript? (Perhaps by submitting it to a journal?)

@DavidZ: I think the question in this case is: why isn't the student showing the manuscript to faculty at his university?  Even if no one there is qualified to read it (which seems unlikely since the topic is combinatorial plane geometry), they can still work with him to find someone who can.

The fact that he compares himself to Ramanujan just gives you all the more reason to ignore his mails.
If his work had any merit, his follow-up email would have been focused on that merit and how it might have been hard to see at first glance.

I don't think that if someone compares himself to Ramanujan, his works become useless. In reviewing what only matters is the work itself. It is immaterial whether he compares himself to Ramanujan or Gauss.

@AlfredGauss Sure, but the fact that his choice of follow-up is that comparison rather than an explanation of the merit of his work says a lot.

but just as the @AlfredGauss worrying about wasting time for a crank, the sender is worrying about being seen as a crank. It's the worry that makes him has to focus on where he is a crank or not, not the content of the paper. And even if he want to say so, how should he replies if there is no question for him?

In fact the tactic of comparing himself to Ramanujan is a well-established logical fallacy: https://rationalwiki.org/wiki/Galileo_gambit

Many years ago, my university used to send letters that explained they received so many proof they didn't have the time to check them all, so each sender received a copy of the previous proof the university had received and asked them to check that one to help the university with their workload. That worked very well.
I think it was my professor in analysis, who received one letter where someone had worked out an excellent approximation of pi as a fraction of rational numbers (I think it was the next approximation better than 355/113). And he found that the result this man found was actually absolutely correct, not quite as mind-blowing as the sender probably hoped, but nevertheless correct, and he replied with a long letter acknowledging the correct results and a list of sources that would help an interested amateur.
That man was a single and outstanding exception. And the OP starter complaining about mistakes: Most of the time things are so bad, there are not even things that could be called "mistakes".

What's special about approximations of pi as fractions of rational numbers? I just found a really good one: 314159265359/100000000000

To be precise, the convergents $a_n/b_n$ of the continued fraction expansion of a given real number $x$ are provably the best rational approximations with denominator $\leq b_n$.

@daviewales: That's actually a rubbish and rather thoughtless approximation, since better approximations with much smaller numbers are known.

OK. I think I misunderstood your meaning. When you said "better than 355/113", I thought you meant "closer to pi", rather than "closer to pi, with small integers and mathematical rigour".

The idea in the second paragraph is excellent!

Have you considered offering your professional services for a nominal fee?  I would think $250-500 that a starting price for detailed analysis, and potential support of a mathematical proof would be a fair price.  Of course for one that will require considerably more effort, that fee could be increased.
If you are loathe to take the money you could always either donate the fees, or return them to the author.  The primary purpose of the fee is to filter out the random amatuer submissions that have not been well thought or checked out.  I assume that you would not mind doing a few serious reviews a year, if you could avoid the spam.

Noah Snyder did this for a bit and made some good money; he tells the story here http://cstheory.stackexchange.com/questions/4489/should-experts-in-tcs-charge-money-to-read-proofs-that-p-np/4508#4508 .

Ramanujan was very poor remember?

@PatrickT - If he was serious about the proof then finding the nominal fee or offering some other service in exchange should not be a problem.  The purpose of the fee is put up a barrier in front those who are not serious that is easy to overcome with a little effort so that those that are serious can get through.

There is a market for this: What I learned as a hired consultant for autodidact physicists

Discuss this with your employer first, you might get into legal trouble.

In a reply, suggest a journal to send it to. Then if it gets accepted for review, the reviewer may have an easy task in front of them. (Either that or genius will be recognised.) Everyone will be happy either way.

Can you suggest some journal to which he may send his approximately-2-page 'elementary proof' of the Sylvester-Gallai Theorem?

@AlfredGauss Hahaha, well, the burden of journal-finding lies with him. He's got access to the Internet. Let me just point out that he may have no idea how peer-review and publishing work in mathematics. Luckily, nowadays you don't have to explain it to them. It suffices to suggest Google search terms.

@AlfredGauss: Possibly this one?

@NateEldredge The source or the length of a proof can't be a basis for its rejection. The AKS paper is only 9 pages http://en.wikipedia.org/wiki/AKS_primality_test

I guess I don't see the big moral quandary. You're a talented person who has worked a long time to develop his skills, and you are under absolutely no obligation to give those skills and your time away for free to every Tom, Dick, and Harriet. If you want to that's fine, but the fact that you feel pressured to do this is not good.
This person's case is really not that compelling to me: he has proved a result which has already been proved and by elementary methods also (according to a previous poster). Maybe it could be published but for this person to suggest his ability is comparable to Ramanujan's in some way based on this result seems absolutely ludicrous. To me, his appeal to Ramanujan, which is based solely on their circumstances and nationality, seems manipulative, and his comparison of himself to Ramanujan, shows a sort of hubris I find appalling. If Ramanujan had sent a proof of a result which had already been proved by elementary means to Hardy, do you really  think Hardy would have given it a second thought?  I seriously doubt it. Based on the information given, maybe he has some talent, but I don't see evidence of a world-class genius being lost here.

The reasoning developed in the second e-mail ("please do not ignore the hidden genius") was true when you first started receiving this type of e-mail, which is why you read these first theorems.
However, after a few try, you figured that the genius/spam ratio (as @EnergyNumbers pointed out) was not worth considering all these e-mail (maybe unconsciously...). In short, I think nothing changed with this e-mail.
If you really want to consider all these emails without spending toot much time, as @bingung said, and if you are giving lectures, you can assign them to students. It would be indeed a great exercice to try to demonstrate the theorems are not valid.
A third option, to give you a good conscience, and since the genius/spam ratio is probably really low, you can review 1/10th of the theorems you receive. It will not dramatically decrease the change to discover a math genius...

After getting rid of spam in your email box by the spam filter, pass every unrequested mail through the crackpot index:
It will give points for, for example:

mentioning Einstein, Feynman or Hawkins. (I suppose mentioning Ramanujan would be the same but in the mathematical field instead of physical field).
complaining about the establishment
vacuous statements

Read everything about it in: http://math.ucr.edu/home/baez/crackpot.html
I don't believe you'll need more then 2-3 minutes for that.
However, also don't forget to take a look at http://en.wikipedia.org/wiki/List_of_amateur_mathematicians, since there is mathematics outside academical mathematics too.

Yes, obviously I meant Einstien, Feynmann and Hankiws, in the case of physics. They are as equally important as Erdoos, Rieman, or Ramaniuya.

This question has been asked more in the mathematical sciences than anywhere else. An interesting text on the topic (with advice) is A Budget of Trisections by Underwood Dudley. It is probably available somewhere cheaper than on amazon dudley (I found some related work on scribd.com.) If you are dealing with an intelligent and younger person, it might be useful to point out that your time is limited and that they might benefit from reading that text. The lesson I learned there is that practically no older amateur will take your advice when it is pointed out that they tried to prove something extremely hard or know-to-be-unprovable. Today this all happens online, and you should also look at John Baez's crackpot index at http://math.ucr.edu/home/baez/crackpot.html , I assume there must be a mathematical version. (translate Einstein to Ramanujan, etc.)

Ramanujan is actually a difficult case because he was in fact an amateur crackpot, and his contribution to real math is not clear to me. But yes, he was a genius. He did not reveal the methods by which he derived his magic formula, even though I am convinced he could explain it if he wanted to. He did not want to reveal his secret craft, he only wanted the fame. I say dunk it in the wastebasket in this case, and if you ever come across work which sounds scientific tell the author to submit it to arxiv.

Referring to arXiv doesn't make sense. If the crank has no academic affiliation, arXiv will need an endorser http://arxiv.org/help/endorsement so the crank will simply return to writing mathematicians to find an endorser.

You show little knowledge of Ramanujan's story.

I think a good policy would be to redirect those people to Math Overflow. Let them open a new topic to ask what's wrong with their proof.
If it's trivial to find errors then someone one Math Overflow will point out those errors. If their proof actually works I would expect someone at Math Overflow to recognise working proof.
You only need to write a email to redirect people to MathOverflow once and afterwards you can send everyone who sends you unsolicited proofs the same canned email.

Please, no. We get plenty of these at MO already, and they get closed immediately because their questions are incoherent.

@DavidSpeyer : If the main reason they get closed is incoherence, the person who asks the question has feedback and could hopefully ask a better question next time

Please don't send amateurs to MathOverflow. This is explicitly not the purpose of MO, so the question will be quickly closed, leaving the amateur unhappy because they were specifically told to post it there. Even aside from the purpose of MO, it can be difficult to convince cranks that there's something wrong with their proof. One critical professional skill is writing clear, readable proofs that are precise and explicit enough that one can point out unambiguous errors if they are present. If someone doesn't yet have that skill, then trying to sort out their manuscript can be an awful mess.

@Christian: Maybe a better suggestion would be to redirect those amateurs at first to MSE rather than to MO. If nobody makes a comment regarding the flaw of the proof then he may wish to post his proof in MO.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Is it worth it for me to try at this point?
Basically, I came into this university on a scholarship, doing their most prestigious undergraduate degree. For a variety of reasons including but not limited to depression and anxiety, I basically failed half my courses in my first year. Last semester, I took a 6 month break.
Right now, I have been kicked from that degree and have lost my scholarship, but I am really enjoying my schoolwork again. A lot of the problems that were plaguing me the past 1.5 years have been largely resolved. I have missed struggling with a difficult problem and/or the feeling I get when I finally understand something. Originally my passion was space and wanted to do something in that area (such as astrophysics). It was (maybe still is) a dream of mine to go into this field at a higher level after undergrad. When I came to university, I was ambitious and wanted to get into some Ivy League school for graduate school, or some top school for whichever field I chose.
I had lost that dream, but now I have found it again. I was wondering if it was worth pursuing and putting all my effort into it, or is it too late for me already with my terrible background?

I'm unclear on the question. Are you asking if you still have a chance at attending an Ivy League school's graduate program or are you asking if you have a chance of a successful academic career?

I think both. Sorry for not making that clear.

Just follow your dreams.

The title of your question seems rather self-defeatist, don't you think? I mean, if your options are to hunker down and overcome a bad start, or to pack it all in and bag groceries for the rest of your life, what do you expect us to say? "Oh, yes, by all means, quit now." Setbacks are setbacks – nothing more.

Voting to reopen.  There is nothing undergraduate-specific in this question.

I agree with JeffE on this one—the question is not undergrad-specific (replace "scholarship" with "fellowship" for the graduate version of this question).

I think this is a simple answer.
Stop thinking and start doing.
If you want a graduate degree at an Ivy League school and a successful academic career (perhaps even in the field of astrophysics), then pull your socks up, set definite, measurable and realistic goals and start achieving them. The path will not be easy but always remember that you are not the first and neither will you be the last.
Note that a graduate degree at an Ivy League school and a successful academic career are not necessarily mutually inclusive. To do one does not mean that you will do the other and neither does it mean that there is a causal relationship between the two.
Personal anecdote:
I had a rather forgettable start to my academic career as well. I recovered enough to get myself admitted to an "Ivy League" graduate program. I am still on my journey and I dare say the destination is years away. Not everyone is a rising star. Some folks are late bloomers.
I reiterate again. To achieve your dreams, just keep trying sincerely and honestly. It will happen in due time.


text:
In what tense (present/past) should papers be written?
That is, should it be present tense or past tense?
Should there be a difference between the abstract, main body and the conclusion?
Does the field of publication have any impact?

I suggest reading a copy of How to Write and Publish a Scientific Paper by Robert A. Day. You could also get a copy of The Elements of Style by William Strunk Jr. and E.B. White.

Don't forget about the future tense (for discussion and open problems).  :-)

The rules of thumb are:

Established facts are reported in the present tense (“The path of light follows Fermat's principle of least time”). However, you should use the past tense when you refer to previous work in the field (“Maxwell et al. demonstrated clearly in a laser cavity experiment that no mirror is perfect”).
The experiments, simulations or calculations you performed are narrated in the past tense (“We dissolved the remaining solid in a 5:1 solution of acetone and benzonitrile, and heated to 200°C for three hours.”)
Discussion of the data presented in the paper uses the present tense (“The results obtained, shown in Fig. 3, clearly emphasize that the cell colonies grew faster on pink toothbrushes than green ones. We attribute this to the color-sensitivity, or kawai factor.”)
Mathematical proofs are written using the present tense, because going through the proof occurs at the time of reading (“From Eqn. 1, we derive the following system of inequalities”).

Overall, the choice of tenses is actually pretty logical.

I'm not sure if I agree with the third example. If the result can be generalised, surely the results show that the cell colonies grow faster, not grew faster, on pink toothbrushes?

@gerrit I agree: here, as written, it is meant to be a specific statement about the experiment (it is “ the cell colonies grew faster”, not “cell colonies grow faster”). Both are of course possible, but have different meanings…

A trickier case is describing mathematical proofs from older papers.  It's "Thurston [4] claimed that 3-manifolds are ..." but "Thurston's argument [4] implies that 3-manifolds are...".

This is a nice succinct description of tense. In the lab report guidelines I wrote for my students, I took two pages.

@JeffE I guess the second usage refers to the use in a proof (and hence present tense) and the first usage refers to some historical context (and hence past tense?).

@Suresh: The subject of the first sentence is a person, who made his claim at a specific time in the past.  The subject of the second sentence is an argument, which implies what it always has and always will, and oh by the way it was first articulated by Thurston.  Platonism FTW!

What about pointing to other parts of the paper: We describe X in section 2 or We will describe X in section 2?

@gerrit passive voice if unsure (“the data described in section 2”)… but my choice would be: past if it refers to a section above, present or future for a section below

@F'x Even with passive is the choise between "The data will be described" or "The data is described". My question is if it should be present or future.

@gerrit I tend to use the present, but I believe both are okay

What if in the Methods section I want to describe a procedure to calculate something, (maybe depicted in a diagram)?. Should I use past tenses because I did it or present tenses because it's something that I did but the procedure always true? For example We calculate the exposure time by taking ... and then we divide....

There are two distinct general cases that bring up the question of, "Which tense should I use?", each of which follows different principles.
First, are you describing research itself and ideas from research? (This is what you are doing probably 99% of the time.) In this case, the principle that I follow is simple, regardless of whether I am describing what other people have done or describing my own work:

Historical occurrences should be in past tense. "Historical" means anything that actually happened in the past, whether a procedure, an article publication, a statement made by anyone, or anything else that has actually happened.
Enduring truths should be in present tense. "Enduring truth" in this context only means that the authors present such statements as ongoing facts in the past, present and future; it doesn't mean anything more than that. (In particular, even if you disagree with what some authors consider to be true, it is still an "enduring truth" in their minds, and so should be presented in present tense.)

The second case where tense is involved is trickier: are you describing your own writing process as you are writing the article? (Although you only do this 1% or less of the time, it leads to perhaps 80% of the confusion of the question as to which tense to use, so it is important to understand this.) The basic idea here is to anticipate your reader's expected or intended path of reading, which is what makes it so tricky. Here is the principle I follow here:

When describing anything you write in the current paragraph or any paragraph below, use present tense. Future tense could also be correctly used for most things in paragraphs below, but not always. In particular, when referring to elements outside the main body of the text (such as appendices, references, footnotes, acknowledgements, etc.), you should always use present tense, since the reader should refer to such ancilliary sections simultaneously with reading the text. I find this a bit confusing, so it is simpler for me to only use present tense and never use future tense, which is perfectly acceptable.
When describing any thing you write in preceding paragraphs, use past tense. The only exception is that when writing in appendices, you should refer to the main body of the text in present tense. (Technically, you should probably also use present tense if referring to the main body in other ancillary sections [references, footnotes, acknowledgements, etc.], but I never need to refer to the main body of the text in such sections.)

I now follow with several annotated examples to illustrate these points. Some of them are straightforward applications of the principles I have summarized above, though some could be argued as either present or past tense.

Literature review
AuthorA and AuthorB (Year) studied an interesting topic. They
  applied a cool methodology and found that certain surprising outcomes are what actually happen.

Comments:

"AuthorA and AuthorB (Year) studied an interesting topic": The study occurred in the past, and so is in past tense.
"They applied a cool methodology": The authors actually did some analysis in the past, and so it is in past tense.
"and found that": The authors arrived at their results at a certain point in time in the past, when they made their conclusions; this is thus in past tense.
"certain surprising outcomes are what actually happen": The authors concluded that their findings not only applied to their study, but indicate some general truths that would continue to apply in the future. Thus, as a description of ongoing reality, it is in present tense.

Methodology Section of My Own Article
We gathered and analyzed certain pertinent data in various stages. First, we conducted an extensive survey. The respondents of the survey reported that they only partially agreed with most of the statements in the survey instrument. Second, we interviewed other relevant respondents. The interviewees enlightened us as to some of the responses on the prior survey. Third, we collected biometric data from the brain electrodes attached to the interviewees during the interviews, which was generally consistent with what they were actually saying (except for some notable discrepancies, which we discuss in more detail below). Finally, we applied some cutting-edge combined quantitative-qualitative analyses to bring everything together and make sense of it all.

Comments:

"We gathered and analyzed", "we conducted", "we interviewed", "we collected", "we applied": Statements of research methods or procedures that we actually did are past events; hence, they are in past tense.
"The respondents of the survey reported", "The interviewees enlightened us", "biometric data ... was generally consistent with": Responses and results (whether by people, animals, plants, or inanimate objects) are reports of historical facts: they actually happened in the past. Thus, they should be in past tense.
"The respondents of the survey ... only partially agreed with most of the statements in the survey instrument", "what they were actually saying": These are tricky. But even when you are reporting what people said to be enduringly true about themselves, their saying such things is a past historical statement. That is, if you were to ask them again today, they might have changed their minds. You can only report what they said as a statement made in the past. This is probably the trickiest point, and might be argued to rather be in present tense, though I personally don't think so.
"notable discrepancies, which we discuss in more detail below": Here you are describing what you have written in the present article, that is, part of your own writing process. This should be in present tense.

Discussion Section of My Own Article
The respondents of our surveys and interviews gave us valuable responses that generally confirmed our hypotheses. We conclude from our analyses that the kind of people in our study generally act in the way that our hypotheses claim.  However, the brain electrode readings give us a more nuanced understanding of our findings. They indicate that people act in that way only in certain circumstances. The appendix of this article provides more details on all this.

Here it gets quite tricky, with cases that might be argued either way. Comments:

"respondents of our surveys and interviews gave us valuable responses": The respondents responded in the past.
"generally confirmed our hypotheses": we conducted the tests of our hypotheses in the past. However, it could be argued to be "generally confirm our hypotheses", since the hypotheses remain confirmed eventoday and in the future by those historical tests.
"We conclude from our analyses": Not only in the past, but now and in the future, each time we reassess our analyses, we continue to arrive at the same conclusions.
"our hypotheses claim": Our hypotheses have not changed; they continue to make the same claims.
"people in our study generally act in the way", "people act in that way": We extrapolate our findings not only to the people who responded in our study, but as a general finding concerning how that type of person continues to act.
"the brain electrode readings give us a more nuanced understanding of our findings", "They indicate that": We are referring here not to the past fact of taking the readings, but to the ongoing fact of our interpretation of those readings.
"The appendix of this article provides more details": This refers to a supplementary section of my own writing.

Appendix of My Own Article
In this appendix we provide more details about the analyses from the study. We conducted even fancier and more experimental analyses to better understand the results. Our supplementary analyses revealed some cases where the general trend did not apply. We explain our interpretation of these supplementary findings in the Discussion section of the article.

Comments:

"In this appendix we provide more details": This is the writing process of the current section of the article.
"We conducted even fancier and more experimental analyses", "Our supplementary analyses revealed some cases": These are procedures conducted in the past.
"the general trend did not apply: This is a past finding from a past analyses. It might be argued to be an interpretation, in which case it should be in present tense.
"We explain our interpretation of these supplementary findings in the Discussion section": This reference to the main body of the article should always be in present tense, since it is written in an appendix.

What if in the Methods section I want to describe a procedure to calculate something, (maybe depicted in a diagram)?.  Should I use past tenses because I did it or present tenses because it's something that I did but the procedure always true?  For example We calculate the exposure time by taking ... and then we divide....

@skan, this would be past tense according to the principles I proposed. You said, "the procedure [is] always true", but that is not the case. In fact, if you were to repeat the study, you would not do exactly the same procedure because the data might change, the subjects might change, and you might use better procedures as you gain more experience. So, whenever you are documenting procedures, you are recording historical information of how you actually did things that one time. That should be reported in past tense.

@skan, you need to provide an actual concrete example otherwise I cannot tell for sure.

Imagine the procedure explains how to compute the minimum value of some example data and the user can also follow the procedure on a diagram. The minimum is always calculated in the same way. Would it be wrong to use the present time?
I'll try to elaborate an example.

For example,  can I write something like this in the methododly? "Let's explain the procedure we followed with an example. Imagine we have three numbers 3, 4 and 5,we have to add the first and second numbers and divide the result by 4. This produces ...."

@skan, in your example, you are not describing the procedure you followed. You are explaining how to carry out the procedure, which is not the same thing. That should be in present tense.


text:
Can I hand out students a few pages from books as supplementary materials?
No textbook is perfect. So if I want to supplement my course with some material (a few pages from different references every now and then, on topics I find lacking or weak in the textbook):

is it legal to make copies of those a few pages and give them to the
students as hand outs?
is it legal to scan them and upload them on
the course site?
should I email the authors to take their permission
1st? what if one author is dead?
in case the above is a violation of copyrights so what should I do then? asking the
students to go read those parts in the library? (the students would need to keep a copy
of the reading/supplementary materials)

Your library will probably have someone who's an expert at this very question

Almost for sure, it depends on country.

In the Netherlands, you can copy around 23 pages without a problem, and without consent of the authors.

First, in most cases, the channels through which you distribute the material to your students (hardcopies, restricted-access course website) is not important. Secondly, the copyright holder for each work is (again, in most cases) the publisher, not the author. Thus, getting permission from the author is not necessary.
So, the surest way to avoid trouble is to secure the permission to reproduce the content from the publisher. Publishers should have an online page (e.g., see here for the American Chemical Society publications) explaining how to obtain this permission. Many academic publishers nowadays rely on a centralized online service called RightsLink, where you can directly select the material you want to reproduce and the conditions in which you will use it:

The tool then tells you if you can get a permission to reproduce at no charge (it usually is if you want to reproduce only small parts, a few figures) or if you would need to pay.
Finally, under US law you may qualify for a fair use right to reproduce parts of a copyrighted content.

For Germany, §53 of UrhG allows actually a lot (compared to other countries' fair use policies). The deal is that flat fees on copying machines, scanners, printers, etc. (as well as on paper) are collected and redistributed to authors.
I find §53 slightly ambiguous for the university teacher:
(3) says rougly:

It is allowed to make copies of small parts of works, of small works, or of single articles that are published or made publicly available in newspapers or journals for personal purposes

to illustrate in teaching at schools, non-commercial facilities for education and advanced training as well as in facilities for professional training in the numbers required for the course participants.
for state exams or exams in schools, universities, non-commercial facilities for education and advanced training as well as in facilities for professional training in the numbers required.

So the "universities" are missing in 1. Usually, I'd say they are covered by those other categories, but they are explicitly listed in 2.
However, for sure the students are allowed to make a copy: (2) 1. runs:

(2) It is allowed to make or have made single copies of a work

for personal scientific use, if and as far as copying is needed for this purpose and does not follow commercial purposes.


text:
Should I abbreviate the author names due to the page limitation?
While I am writing the references for my paper, I usually copy and paste the bibliographic information and use it directly. AFAIK, most of them are in the full version, meaning that all the author names are spelled out, e.g. James Rhianna Smith. (They can be really long!).
Also given the fact that some conferences do have a page limit (e.g. 6 pages maximum), directly using those tedious full names is sometimes a luxury I cannot afford.
Under this type of circumstances, should I shorten the author names?

@PeterJansson I don't really agree.  This isn't a technical question about BibTeX, but one about acceptable bibliographical practices.

I've edited the question into a software-agnostic question about bibliographic practices

At the risk of being off-topic, note that even if you decide to use only first initials, you should not need to manually edit your BibTeX file.  Instead, choose a bibliography style that uses initials; BibTeX will abbreviate them automatically.

@NateEldredge I don't think it is offtopic. The OP clearly doesn't know how to use BibTex properly. The answer to both the original and edited question is a simple: "NO!", and has nothing to do with page limits or conference formatting standards.

This is almost always dictated by the style guide of the journal/conference/whatever to which you're submitting. Check with them, or better yet, use a bibtex style definition file provided by them (e.g., IEEE).

Sure, why not?  There's nothing wrong with using first initials consistently.

Depends on what (journal/thesis/conference/etc) style you're following…

And what if two different authors have the same initials and last name?

Well, and what if the full names are the same? This is merely an aggravation of an existing problem, not a new one, so whatever techniques you used before remain valid now as well.

I just experimented with BibTeX and was surprised to find out that it treats authors with the same initials and last name as the same person even if the names have been spelled out.  That is, it will treat Joel Reyes Noche and James Randi Noche as the same person.  This means that my comment above is moot.


text:
Supervisor and GRA roles in submitting a grant
I am required to complete 300 practicum hours to fulfill my requirements in a Masters program.  I decided to fulfill these hours by creating a practicum experience with my GRA (graduate research assistant) supervisor.  This experience was designed for me to develop skills in tasks that I would not normally do in my GRA.  We decided to work on a few different projects- I was to submit an letter of intent after taking a course on grant writing, help develop a couple of publications in the works, and work on a presentation for our group.
I took the grant writing course, developed some general ideas with my supervisor on paths that we might pursue, and we submitted the grant.  I was told that I would take the lead on this grant as part of my experience.  And I have for the most part.  We did a lot of editing together, we met with the grant coordinator of our school, and I did most of the research for the grant.  It was a continuation of an earlier project, but the application was novel and it was something that I created and took ownership over.  I was excited as this was my first grant and I worked very hard on it.  I wanted this to happen.
Months go by and we are working on the other parts of the practicum, mostly working on some publication work.  But it felt like I was being excluded from the main discussion and I was not receiving much support.  It is an area that is not in my background, but I wanted to get experience in this area.  It felt like I was wasting hours sitting there with no support, so I did not count my hours towards the practicum.  I would try to get her to help, but it always seemed like I was bothering her or she didn't have time or was frustrated.  When I told her that I have not counted my hours recently, that I was just doing GRA hours, she seemed frustrated and told me that I need to get this over with.  I didn't think I was getting enough out of the experience.  At this point I started regretting doing my practicum experience here because I would have had a chance to develop real skills elsewhere.  Then, I thought that I probably should have been given more tasks in my GRA hours.  I was horribly underworked to begin with, and my practicum was an attempt to gain more meaningful experiences out of this group.
Recently, we get the word back that the letter of intent was accepted and there was a very good chance that we would get the grant.  I reply back to the grant coordinator like I had done before, since I was taking the lead on this grant, and instead, my supervisor gets in and starts coordinating things.  I keep trying to set up meetings and figure out the next step, but she took over the project.  No discussion, no email at all.  Now there is a lot of work going on with this grant and I feel excluded.  She has not assigned me a task to work on for this and I have not heard anything.  I feel like I am left in the dark.  I know she has a lot more experience and skill with these things, but this feels like it would be a big opportunity for me.  To be honest, I would like some recognition from department.  I feel like I helped the department score funds with a practicum and grant idea that was mine.  I feel pushed away on both the grants and the publications.
Can you think of a way to approach this?  I really do like her and I don't want to cause any issues.  I just think she is a poor leader and supervisor.  I viewed it as a team effort to begin with and I was hoping to continue this way, but it does not feel like it is going this direction.

So what did she say when you asked her about this directly? (You did ask her about this directly, didn't you?)

You seem to be having two issues. The first is you are not happy with your GRA experience and the second is you are confused about the grant. It is important to separate these issues when talking to the supervisor and only bring one issue up at a time. Sicne you questions focuses on the grant, so does my answer.
While it is too late now, before starting to work on a grant or any research project it is important to define the roles of each team member. You then need to review these roles frequently as the project progresses.
There are two possibilities for the grant. The first case is that the grant includes funding for you (or someone like you). In this case you need to have a friendly chat to discuss what your role will be if the grant gets funded. This should be fairly painless since you (or someone like you) is listed on the grant. The second case is that the grant doesn't include funding for you (or someone like you). If this is the case, then it is pretty clear that you are not really included on the grant. If the grant could have included funding for you, but you were not included, you should have raised the issue as soon as you realized. This is potentially problematic and it is worth discussing with the supervisor why you were not included. If the grant could not have included funding for you, then it is not clear what the issue is.
Whatever happens, you should have a nice civil conversation with the supervisor.


text:
Mean life of Universities
I was just thinking, universities (as a public institution) are one of the most long-lived institutions in a human society. The may survive changes of government
and they evolutionate with the community they belong.
Is there any serious study that somehow correlates the duration of a university and a style of education with the society it belongs?
What are the most ancient universities that are still active?

http://en.wikipedia.org/wiki/List_of_oldest_universities_in_continuous_operation

The data we have so far is consistent with the hypothesis that the mean lifetime of universities is infinite.

Here is a book chapter on this topic:
Mazza, C., Riccaboni, A., & Quattrone, P. (2008). Found in translation: On the persistence of the university as institution. European universities in transition: Issues, models and cases. Northampton: Edward Elgar Publishing.
These authors argue that the longevity of traditional universities is due to their socially legitimated role to "make ideas, people, and disciplines meet and cross-fertilize".  For centuries, this role was manifested in such artifacts as great libraries (and later university presses) and degrees for the elites in government, religion, and (to a lesser extent) business.  Of course, being self-sustaining financially (endowments) and physically (buildings and infrastructure) has been an important factor.
Historically, the university was the first institution in human history to systematically question itself as part of it's very constitution.  From the start, the communities of scholars engaged in systematic inquiry and critical debate on the subjects of study, and even the way those subjects were investigated and taught.
Though far from perfect, this self-critical capability gave universities the capacity to take in and develop new knowledge and to let go of knowledge systems that no longer had intellectual support.  For most of their histories, universities shaped the educational norms of the society around them -- at the national and international elite levels, at least -- rather than being shaped by those elites.  The certainly weren't responding to the changing educational preferences or needs of the general public.  Other institutions filled in this role (gap), including public universities, land-grant colleges, community colleges, polytechnic institutes, and so on.

The oldest universities in continuous operation are:

University of Bologna - 1088
Oxford University - 1167
Cambridge University - 1209
University of Salamanca (Spain) - 1218
University of Padua - 1220
University of Naples Federico II - 1224 (first public
university)


text:
How do I get out of my problem of not achieving results, feelings of worthlessness, guilt and rejection?
I am currently a PhD student in a biomedical related field.
I came a long way. I first did my undergraduate and worked for a few years, though not research related. Later, I identified my research interest before embarking in the PhD with the hope of fulfilling my career dreams. I used to love and enjoy science, but that was long ago. I was happy when I got into the PhD program. Now I'm nearly 2 years into my study, mostly 'wet' lab and basic sciences research.
Unfortunately, my experiments - most of them - fail. Initially, that was due to lack of skills and experience - with time things have improved but I am still lacking good results that can give clear direction in my work.
My project has been modified a few times because initially there were problems with the biological samples so I had no choice. Then, I tried to 'reproduce' a piece of result done by a colleague few years ago but the results were always 'off' no matter how many times I did it. Even someone else with better experience in the lab tried but could not replicate the same result. The conclusion was perhaps, change in phenotype of biological samples over time.
Then my project changed again, and the technique is now becoming more complicated and 'taxing'. Partly, because my supervisor is not very happy and feels that it is time I should be producing good data and should aim for higher impact. With more complicated experiments, I also need to beef up my basic sciences knowledge but as someone coming from a different background it is taking me extra time to grasp. With my confidence going in a downward spiral.
I am finding more and more difficult to articulate my thoughts clearly to my peers and I feel constant 'rejection' and 'guilt'. Most of my colleagues are actually good people, but harsh words, although not ill-meaning, are slowly eroding my confidence. When I ask questions because there is something that I don't fully understand (without prior knowledge and not something I can read up on), people appear impatient and ask back (e.g. "why are you asking?",  "what do you think?", or just "I can't give an answer to you on this"). It feels more and more difficult for me to ask questions or speak up for myself.
One day, I was blamed for something (from someone outside the lab) that I didn't even do. The accusation was pretty unreasonable - others who witnessed the event felt the same - but to me I felt I was just a bully target. I just broke down and cried because I realized I couldn't take the stress and frustration of rejections and failure.
My friend in the same lab gave me some counseling but also at the same time asked me whether working in this lab is truly what I want to do. I don't want to quit my PhD, but then again I don't know how I can get out of this situation.

Should I change project?
But my PhD qualifying exam is coming up and I am having a massive problem with writing up a good proposal because of all the frustrations happening.
Should I change to dry lab?
I am afraid of talking to my mentor because I don't know whether at my current mental state I'll just suddenly break down again.

In summary, my resentment and contradictory statements that I receive:

Ideology: To do a PhD, perseverance is more important than being smart.
Reality: You need to be smart. REALLY SMART.
Ideology: There's not such thing as a stupid question. Ask if you don't understand.
Reality: Huh? Why do you even ask?

Hi, and welcome to Academia.SE!  Your question is very complex and specific, but I think you may find the answers in the linked question about dealing with discouragement helpful.

I want to draw attention that in many times results are just results, whether they are good or bad. Some papers report failures, the contribution would then be saving others the time to try this approach. Someone could cite a failed experiment saying something like: "Approach X was shown to be not suitable for achieving Y, so we investigated approach Z".

You should really consider talking to a mental health professional before you make any decisions. I really feel like you might be dealing with depression.

Should I change project?
Should I change to dry lab?

A PhD can be extremely frustrating and discouraging.  Whatever you do, don't make a major decision while you are stressed and upset. Take a deep breath (and maybe a day off) and decide what you need to focus on.
Since you will be qualing soon, you likely need to spend a significant amount of time studying for that.  Passing your quals are important, and will likely give you a confidence boost.
Secondly, consider finding a social outlet.  Many grad departments have a weekly social meet-up at a bar.  Apart from meeting people that work in different labs, it is also quite likely you will meet other people who have gone through this issue, and will help you.

It's good that you are reaching out for help. To address all the issues and problems you face will be a long process. Start by focusing in just the first steps.
First, I suggest you recruit a "coach" -- a respected peer, a recent graduate, a researcher from another department, or a PhD who works in industry.  This "coach" should not be your supervisor or anyone in your lab.  You need someone with whom you can be totally open and honest, and even be "messy".  This "coach" won't give you much advice and certainly won't solve any of your problems.  They just need to be understanding, supportive, and consistent in their connection with you.
Talk with your coach every day for at least 15 minutes.  Set short-term goals -- just for the next day -- and hold yourself accountable in front of your coach.  Set a few number of small goals, e.g. "Tomorrow, I want to rewrite four pages in my lab notes".
Second suggestion -- start a journal (NOT on a computer or on the Internet) where you record ONE positive experience or accomplishment every day, no matter how small.  Maybe it is just something you noticed or someone you appreciated.  Maybe it is something that made you laugh.  The point of this is to make a habit of paying attention to -- and soak in -- positive and expansive experiences.
Again, these two suggestions are just the first steps, aimed at giving you a more solid footing emotionally so you can deal with the bigger long term challenges you face.


text:
How to ask a professor to join a project in a way that leaves him a "graceful out"?
 This is a follow up question to a previous question I have asked about co advising :  professor will agree to be my thesis advisor but only if I can find a co-advisor  . After sitting in on this professors lab meetings I would like to work on two projects that he is getting off the ground in collaboration with another professors lab. How can I ask him in a way that gives him a graceful out even though he agreed to co-advise on my master's thesis? I would really like to work with him since I am really interested in his research, but I do not want to end up trapping either of us in an advising relationship that would be bad for both of us in the event he was just trying to be nice when he agreed to be co-advisor on my master's thesis.

Pretend he's a human being, and then ask him directly.

I do not want to end up trapping either of us in an advising relationship that would be bad for both of us in the event he was just trying to be nice when he agreed to be co-advisor on my master's thesis.

Give him some credit and don't assume he would agree to a bad advising relationship just to be "nice."
Ask him directly. If he wants to say "no," there are any number of ways in which he can say it.

If he's made it to professor, you can be sure he knows how to say no, gracefully or otherwise. :)


text:
Publishing a literature review
Over the course or the past year, I've become very familiar with a specific topic through a careful and systematic review of the background and state-of-the-art.
I think this could make for a nice publication, providing readers in similar fields with a handy resource and increasing my academic profile to boot.
However, I want to make sure that the lit review I performed is scientifically sound - so it must be transparent and replicable.
Do you know of any guidelines, best practices, or great examples to guide me as I get working?

What is your field?  I would think that practices could vary widely.  In your field, do literature reviews like this routinely get published on their own?

Human ecology. Yes, I've seen a few.

While it is not specifically for general literature reviews, but more aimed at systematic reviews and meta-analysis, the PRISMA Statement is widely used in the biomedical literature, and even comes with checklists to make sure a manuscript has a few key elements, such as:
"Present full electronic search strategy for at least one database, including any limits used, such that it could be repeated."
Which is something that's pretty necessary for repeatability - you will often find for systematic reviews that an appendix is included with the actual queries used, what dates, etc. In a more general sense, that's what I'd look toward: "Does my review follow most of the guidelines for presenting a meta-analysis, even if it doesn't bother with summary estimates?" If you do that, you're likely on pretty solid ground.

I would suggest you to perform a limited search for dedicated literature review (survey) papers, either in your field (not sure, whether this should be preferred, as it IMHO might push you too far outside of your own "big picture" view of the field), or in any other field that you can digest.
In addition to the above, please check my recent and relevant answer to a similar question here on this site - along with some general advice, it contains some useful references on the topic: https://academia.stackexchange.com/a/45290/12391.