Questions tagged [soft-question]
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
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Publishing conjectures
One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was ...
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Discovered Phd topic has already been worked on
I am a second-year French Ph.D. student and two days ago I found out the topic I had been working on has already been studied, and the result I wanted to prove is basically already known. ...
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Synthetic vs. classical differential geometry
To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
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Why "open immersion" rather than "open embedding"?
When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
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What are some ways to stay engaged with the mathematical community from outside academia?
I will be graduating with a Masters degree soon in mathematics. For various reasons I have decided not to pursue a career in academia for now and will instead be working a job in industry that will ...
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Status of PL topology
I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
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Are there any "related rates" calculus problems that don't feel contrived?
I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...
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How closed-form conjectures are made?
Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ ...
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What to do if you notice a substantial improvement to a result in a paper whilst refereeing it?
What would you do/have you done in such a situation?
Hand out the improvement for free in your report
Wait until the result is published and then submit elsewhere
Inform the editor about the ...
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Prominent non-mathematical work of mathematicians
First of all, sorry if this post is not appropriate for this forum.
I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things.
Recently I ...
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How to respond to "I was never much good at maths at school." [closed]
We've all heard it. I even got it in Norwegian recently. It's number 1 on the list of responses to the statement "I'm a mathematician.". Does anyone have any good comebacks? What other responses ...
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What areas of pure mathematics research are best for a post-PhD transition to industry?
I have a student who is looking to start a PhD in pure mathematics. She is talented and motivated, and will do quite well. She is still in a phase of her development where she is still open to the ...
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In what respect are univalent foundations "better" than set theory?
It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).
Part of what makes ST so appealing ...
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Is it possible to have a research career while checking the proof of every theorem that you cite?
A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless they had checked the proof of every theorem ...
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Examples of conjectures that were widely believed to be true but later proved false
It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?
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What are some mathematical sculptures?
Either intentionally or unintentionally.
Include location and sculptor, if known.
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Cocktail party math [closed]
Ok, hotshots. You're at a party, and you're chatting with some non-mathematicians. You tell them that you're a mathematician, and then they ask you to elaborate a bit on what you study, or they ask ...
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What about a mathematics journal for 'negative' results?
In the empirical sciences, there are a number of journals that publish 'negative' results. Negative or null results occur when researchers are unable to confirm the findings obtained from earlier ...
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What are examples of books which teach the practice of mathematics?
One may classify the types of mathematics books written for students into two groups: books which merely teach mathematics (i.e., they present theorems and proofs, ready-made, as it were) and those ...
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How to make Ext and Tor constructive?
EDIT: This post was substantially modified with the help of the comments and answers. Thank you!
Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most non-...
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There must be a good introductory numerical analysis course out there!
Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant ...
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Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
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Nontrivial question about Fibonacci numbers?
I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course.
Here is a (not so good) example of the sort ...
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Why is it useful to study vector bundles?
I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...
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Books/websites which have motivating stories of mathematicians overcoming hardships in life
Edit 1: I have received a lot of great answers. I am not accepting any answer because I think there might be in future that some user want to contribute any new answer, as in my opinion some users ...
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Capitalization of theorem names
I hope this question is suitable; this problem always bugs me. It is an issue of mathematical orthography.
It is good praxis, recommended in various essays on mathematical writing, to capitalize ...
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A clear map of mathematical approaches to Artificial Intelligence
I have recently become interested in Machine Learning and AI as a student of theoretical physics and mathematics, and have gone through some of the recommended resources dealing with statistical ...
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Homological Algebra texts
I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe).
As usual, ...
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Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...
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Work of plenary speakers at ICM 2018
The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from 2014 and 2010. Can you, please, ...
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Memorizing theorems [closed]
I always have trouble memorizing theorems. Does anybody have any good tips?
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How can an extremely mathematically talented young person be helped to fulfill his/her potential?
Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented $11$ years old student and I don't know how I can help him. For ...
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Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. ...
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How often do people read the work that they cite?
I have the following question:
How likely it is that an author carefully read through a paper cited by him?
Not everyone reads through everything that they have cited. Sometimes, if one wants to ...
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Why differential forms are important?
Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...
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Advice for pure-math Phd students [closed]
Pursuing a Phd in pure math can be a daunting task. A number of students who begin a Phd in pure math don't complete it, and there are high-quality dissertations and those which are not so high ...
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On referee-author communications
Every time I referee a paper, I dream of a system which would allow me to ask the author a question without troubling the editors. It would save time for everyone involved, most importanly the referee;...
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Request for examples: verifying vs understanding proofs
My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
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What is the etymology of the term "perverse sheaf"?
Grothendieck famously objected to the term "perverse sheaf" in Récoltes et Semailles, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living ...
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Why are the sporadic simple groups HUGE?
I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them ...
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Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
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Why do we need random variables?
In this MathStackExchange post the question in the title was asked without much outcome, I feel.
Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not ...
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Publication rates in Mathematics
Have there been any studies of publication rates in Mathematics?
We are trying to construct a workload model for the Faculty of Science and Engineering at my institution. Part of this involves ...
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Most intriguing mathematical epigraphs
Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing.
To pick up an epigraph is some kind of nearly mathematical problem: it ...
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Undergraduate math research
I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level.
I was recently accepted into the McNair Scholars program. It ...
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Changes forced by the pandemic
The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
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What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
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Modern results that are widely known, yet which at the time were ignored, not accepted or criticized
What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It ...
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Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number ...
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How does a mathematician choose on which problem to work?
Main question:
How does a mathematician choose on which problem to work?
An example approach to framing one's answer:
What is a mathematical problem - big or small - that you solved or are ...