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As a sequel to my previous post “ Redundancy of letters of recommendation for scientific and mathematical majors”, I wish to ask a relevant question.

I have been informed that a master’s student of mathematics is not required to have published a paper in order to be admitted to a doctoral institution. However, my applications all experienced rejection so I am trying to write my first paper.

However, my knowledge in mathematics is limited and in a limited time I cannot come up with an impressive result. On the other hand proving a small result and having it published is against my scientific principles, but I think I will finally do it.

Is it possible that a high gpa (19.5 out of 20), a simple published mathematical paper and a good TOEFL grade, but not good letters of recommendation or lack of letters of recommendation will guarantee admission to a top-tier university?

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    your title probably needs to change. – Boaty Mcboatface Jan 1 at 14:28
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    yes, but it was just a small survey of a theorem rather than coming up with an original result @scaaahu – user117892 Jan 1 at 14:48
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    Did you have an advisor when writing your master’s thesis? Normally this person would feel responsible for writing you a letter of recommendation. If they didn’t agree, it suggests that something more is going on than you being introverted. Instead of trying to work around the system by writing a math paper when you don’t have the relevant skills or an idea that you consider worthy of publishing, my suggestion is to confront the problem head on and try to understand why these professors didn’t want to write you a letter. – Dan Romik Jan 1 at 15:37
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    It is more likely that your lack of recommendation letters is hurting you as much or more than your lack of a publication. – Morgan Rodgers Jan 1 at 16:31
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    @Mathephile2020 I think this tells you, you should be working with a professor or two to help write your paper and build relationships that will get you better letters. – Morgan Rodgers Jan 1 at 17:25
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Nothing guarantees admission at top-tier universities. Sorry. It doesn't work like that. The competition is fierce.

But note that the expectations for entry into any doctoral program vary by country. In the US, a doctoral program normally starts with coursework leading to the comprehensive exams. So less (in the CV) is likely to be required on entry. Other places, more is needed.

Again, though, you really need good letters of recommendation. I don't expect much success otherwise.

As to publishing, small results will probably help everywhere, even if not dramatically, but you might be able to do something bigger if you do it collaboratively. And, especially, if you can do a publishable project with one of your professors you can hit two targets simultaneously; both a better publication and a good LoR. Even a paper more or less ready for publication along with a good letter would be a plus.

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To add to @Buffy’s answer, in all math graduate programs I’m aware of, at least in the US, letters of recommendation are a requirement. So a “lack of letters of recommendation” is actually a guarantee that you will not get admitted. This is not something you can solve by publishing either one “simple” paper, or even many non-simple ones, except in the indirect sense that if you publish many papers then good letters of recommendation are likely to follow more or less automatically.

Keep in mind that this applies to all programs, not just “top-tier” ones.

  • This requirement needs to be reviewed and modified if not canceled – user117892 Jan 1 at 15:35
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    @Mathephile2020 telling all of academia that it needs to change, from the perspective of someone who doesn’t even understand why academia works the way it does, isn’t helpful and doesn’t show a great level of maturity on your part. Perhaps it is your own ideas that need to be reviewed and modified. – Dan Romik Jan 1 at 15:40
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    There is a social aspect to university studies - the elusive collegiality - that one cannot ignore. Nobody wants to teach or work with a highly abrasive student. – ZeroTheHero Jan 1 at 15:48
  • I think what I meant wasn’t got across well. In my opinion, even one letter of recommendation from a supervisor is enough while from more than one referee might be considered as optional but favorable. – user117892 Jan 1 at 16:12
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    @Mathephile2020 thanks for the clarification. There isn’t room here to discuss why the system works the way it does, and I don’t claim that it’s a perfect system either and cannot possibly be improved, but that’s beside the point. If you want to get into grad school you’ll have to play the game by its current rules. Complaining that you don’t like the rules won’t get you anywhere. Anyway, good luck! – Dan Romik Jan 1 at 16:25
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I cannot imagine getting in a top program without reasonable letters of recommendation, and I see this as much more important than having a published paper especially as in a previous post you wrote “ However, the only issue in my doctoral applications is apparently the loathness of almost all of the university teachers who know me to write a letter of recommendation for me”.

You also admit your knowledge of math is limited: indeed without a strong GRE math score, a high gpa is not very meaningful.

You need to realize there are finitely many slots available and that pretty much all those admitted to a top program will have high gpa but will also have strong reference letters. A paper might help but with your limited knowledge and without a mentor to help you putting a result - even an average result - in proper shape, the odds of publishing in a reasonable journal in a fairly short time are next to nil.

  • By a limited knowledge I meant I had still a long way to publish a paper at a top-tier journal. I think GRE is all about calculus and basic algebra if I’m not wrong – user117892 Jan 1 at 15:48
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    Take the GRE and ace it then. – ZeroTheHero Jan 1 at 15:49
  • My next exam is TOEFL. I would like to take a GRE test too – user117892 Jan 1 at 16:20
  • The GRE has an advanced subject matter test that is pretty challenging. It tries to account for the wide variety of programs and so covers ground that many students have never been exposed to. I was totally depressed after finishing the exam (many years ago), but wound up doing pretty well. Not very school requires it, but some do. – Buffy Jan 1 at 16:39
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I am pretty sure the answer is no. There are very few things that can guarantee admission to a top tier school. Some of the things that can get near guarantee a the top of my head,

  • A faculty willing to work with you.
  • A very strong research background that admission board can recognize its strength.
  • Big dollar donations (this is true of undergrad admissions, not exactly sure if it would work in grad. level)

About the last bit, an ex-undergrad. admission official said on camera that it would be 6-7 or even 8 figure donations but sometimes 6 figures + a connection from the school board would do it. I can fish the source if anyone wants it.

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    @buffy nope, I am pretty sure it is legal. But yes, I am probably being cynical. I agree it is unethical. There are very shocking testimonies about this issue regarding UG admissions. – Boaty Mcboatface Jan 1 at 14:32
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    @SolarMike, hmmm, The Dr. Buffy Center for Mathematical Studies. Nope, doesn't work. – Buffy Jan 1 at 14:47
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    @Buffy, I sometimes wonder if people ever downvote things they do not wish to believe it could ever be true. – Boaty Mcboatface Jan 1 at 16:31
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    @BoatyMcboatface do you really think it’s helpful to OP, and to the world at large, to give out practical information, including links to videos, on how making unethical use of large amounts of money can get you ahead in academia? I agree with @Buffy’s critique, and suggest deleting that part of your answer. – Dan Romik Jan 1 at 16:31
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    @DanRomik, deleting comments as this only empowers the ones who are involved in this practice. People who have this sort of financial power are either aware or easily made aware of this possibility. On the other hand, no one really talked about this way of getting into colleges until recent admission scandals. It is for the benefit of us all to understand what "big school" tags entail. If do not openly dispute this system and question the reputation of these "top schools" then who do we get ahead of this? – Boaty Mcboatface Jan 1 at 16:37