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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60 votes
9 answers
5k views

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 ...
60 votes
6 answers
8k views

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. ...
60 votes
6 answers
10k views

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 ...
ಠ_ಠ's user avatar
  • 5,933
60 votes
1 answer
6k views

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 ...
59 votes
11 answers
6k views

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 ...
59 votes
7 answers
7k views

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. ...
59 votes
5 answers
24k views

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 ...
59 votes
7 answers
4k views

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 $Y_\mu(x)$ ...
Vladimir Reshetnikov's user avatar
59 votes
7 answers
7k views

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 ...
58 votes
82 answers
18k views

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 ...
58 votes
22 answers
14k views

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 ...
58 votes
12 answers
33k views

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 ...
58 votes
7 answers
8k views

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 ...
58 votes
4 answers
7k views

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 ...
57 votes
30 answers
16k views

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?
57 votes
43 answers
11k views

What are some mathematical sculptures?

Either intentionally or unintentionally. Include location and sculptor, if known.
57 votes
25 answers
11k views

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 ...
57 votes
5 answers
6k views

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 ...
57 votes
16 answers
8k views

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 ...
57 votes
5 answers
8k views

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-...
darij grinberg's user avatar
57 votes
8 answers
16k views

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 ...
57 votes
4 answers
5k views

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 ...
56 votes
28 answers
11k views

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 ...
56 votes
12 answers
14k views

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 ...
user avatar
56 votes
10 answers
4k views

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 ...
56 votes
7 answers
16k views

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 ...
56 votes
10 answers
7k views

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 ...
56 votes
12 answers
27k views

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, ...
56 votes
4 answers
12k views

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 ...
Martin Brandenburg's user avatar
56 votes
3 answers
11k views

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, ...
55 votes
19 answers
46k views

Memorizing theorems [closed]

I always have trouble memorizing theorems. Does anybody have any good tips?
55 votes
18 answers
8k views

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 ...
55 votes
9 answers
6k views

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. ...
55 votes
10 answers
6k views

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 ...
55 votes
10 answers
18k views

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 ...
55 votes
5 answers
48k views

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 ...
55 votes
7 answers
3k views

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;...
54 votes
15 answers
5k views

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 ...
54 votes
6 answers
12k views

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 ...
Daniel Litt's user avatar
  • 22.2k
54 votes
5 answers
10k views

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 ...
REDace0's user avatar
  • 677
54 votes
12 answers
4k views

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 ...
54 votes
16 answers
15k views

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 ...
Filippo Alberto Edoardo's user avatar
54 votes
6 answers
15k views

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 ...
53 votes
34 answers
14k views

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 ...
53 votes
16 answers
66k views

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 ...
53 votes
10 answers
7k views

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, ...
53 votes
11 answers
6k views

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 ...
53 votes
14 answers
9k views

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 ...
53 votes
6 answers
5k views

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 ...
53 votes
9 answers
11k views

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 ...

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