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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5
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0answers
129 views

If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?

This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general? For instance, McMullen's polytope algebra is ...
49
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65answers
13k 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 ...
12
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3answers
983 views

New research on coding in reverse mathematics?

Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey ...
1
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0answers
129 views

Online seminars [duplicate]

In the time of COVID-19 everything went online including seminars. A good thing about online seminars is that they can be attended by everyone and many seminars that I attended had more than 50 ...
40
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12answers
7k views

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
21
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10answers
5k views

Best Springer mathematics books [closed]

I received an email from zbMATH today to the effect that, as a result of their going open access in January 2021 (which is awesome), they will no longer be able to offer their reviewer discount for ...
141
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25answers
37k views

What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
49
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4answers
5k views

A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?

Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power. Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...
23
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0answers
714 views

Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory ?)

The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...
2
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0answers
103 views

A taxonomy of proof methods [closed]

I am looking for a taxonomy of proof methods in mathematics. For basic proof methods I would think of proof by contradiction, mathematical induction, structural induction (yes I am a computer ...
61
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15answers
15k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in most ...
30
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3answers
1k views

Wiki for consequences of axiom of choice?

I raised the following question as part of another MO question, but I am following the suggestion of Nate Eldredge to make it a question in its own right. For many years, there has a been a valuable ...
193
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42answers
71k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
75
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10answers
9k views

Least collaborative mathematician

The recent question about the most prolific collaboration interested me. How about this question in the opposite direction, then: can anyone beat, amongst contemporary mathematicians, the example of ...
167
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61answers
80k views

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...
136
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50answers
17k views

Old books you would like to have reprinted with high-quality typesetting

There are some questions on mathoverflow such as What out-of-print books would you like to see re-printed? Old books still used with answers that tell us things such as: Mathematicians prefer to ...
31
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2answers
1k views

Wayback Machine for mathematics?

I have had a couple of experiences recently which have made me wonder whether the mathematics community should try to establish and maintain something like the Wayback Machine, but specifically ...
25
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0answers
2k views

What do you like in the mathematics of Vaughan Jones? And how Vaughan Jones liked mathematics to be? [closed]

Edit: Directed to mathematicians ,I think this is a suitable time to open this question to know more about the creative mathematician Vaughan Jones Probably his students are ready for getting answers ...
18
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2answers
1k views

Is it acceptable to use the citation references like [1] or [Joh] as nouns in mathematical writing?

I have seen this enough times that I thought it was common practice, but I recently saw someone on a website about writing tips (can't find the link now) say that this was not good. I am hoping to get ...
10
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0answers
359 views

Recommendations on PhD programs in Europe on number theory

I am a Master student of math in the UK, and I would like to do a PhD in number theory, preferably the algebraic side of it. Given that I am in the UK, I would probably prefer staying here or going to ...
11
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1answer
879 views

Treating citation numbers as objects [duplicate]

The following item appears in the $\LaTeX$ style-guide for the Journal of Integer Sequences: To me this seems correct and reasonable, but it in most articles I read, authors tend toward the first ...
32
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6answers
2k views

Does seeing beyond the course you teach matter? The case of linear algebra and matrices

This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
126
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18answers
9k views

Suggestions for special lectures at next ICM

(I am posting this in my capacity as chair of the ICM programme committee.) ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's ...
97
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9answers
28k views

How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...
57
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17answers
11k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
33
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5answers
4k views

How to deal with an advisor that offers you nearly no advising at all?

I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely. ...
6
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3answers
2k views

Can American Math. Monthly be used to publish hard research?

My question pertains to the journal "American Mathematical monthly" published by the MAA. I wish to ask whether a paper as a part of a PhD thesis (subject: Combinatorics ) can be submitted ...
70
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13answers
18k views

How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...
32
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5answers
3k views

Advice for researchers outside academia

Perhaps some personal background is relevant to this question. A couple of years ago, I graduated with a master's degree in Applied Mathematics from a good Dutch university. Even though I obtained ...
13
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9answers
2k views

Tools from other disciplines useful to mathematics research?

Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering ...
24
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2answers
705 views

Sustainability of ZBmath unrestricted access

ZBmath (formerly Zentralblatt für Mathematik) will become "Open access" in 2021 (see for instance at EMS site and at FIZ Karlsruhe site). It used to be under paywall, although some partial ...
3
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1answer
382 views

Heuristics for boundary Harnack inequality

What is the heuristic idea of the proof of the boundary Harnack inequality presented in the appendix of Caffarelli's 1998 lectures on the obstacle problem (page 38 here)?
57
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21answers
8k views

Small ideas that became big

I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research ...
3
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0answers
408 views

Has anyone discovered a flaw in a published proof and no one wants to rectify the situation? [closed]

A post appeared yesterday (9/22/2020) entitled ``Fear of publishing a wrong result'' on the Mathematics Stack Exchange. https://math.stackexchange.com/questions/3836536/fear-of-publishing-a-wrong-...
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4answers
1k views

Which mathematics book would you wish theoretical physicists had read?

It is a well-known phenomenon that mathematicians and physicists working on closely-related topics (say in gauge theories, or in general relativity etc.) generally approach the problems from very ...
2
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0answers
95 views

Frobenius reciprocities

An adjunction of the form $\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies ...
302
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47answers
92k views

Widely accepted mathematical results that were later shown to be wrong?

Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly ...
3
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2answers
606 views

What are your opinions on Zeidler's QFT books? [closed]

I am interested in mathematically rigorous treatment of quantum field theory, constructive QFT in particular. I have read 'QFT, A Tourist Guide for Mathematicians' and am going to read "Quantum ...
35
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1answer
3k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov–Witten invariants that satisfies the Korteweg–de Vries hierarchy. Let me state the fact more precisely. ...
105
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20answers
17k views

Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public. Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...
14
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4answers
893 views

Examples of (Git) open math (texts) projects

I am an active part of a research project on the positive effects of open math projects on the community. With open math projects I have in mind a particular thing, namely a GIT project on mathematics ...
29
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4answers
2k views

Online, evolving, collaborative foundational text projects

There are two online, evolving, collaborative "foundational text" projects for research mathematicians that I am aware of: (1) The Stacks Project for algebraic geometry (2) Kerodon for ...
2
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0answers
336 views

Reference request for Miyaoka's wrong proof of FLT [duplicate]

Before Wiles and Taylor's proof of Fermat's Last Theorem, Yoichi Miyaoka famously (and wrongly) claimed to have solved the problem, at least for sufficiently large prime exponents. However, i have ...
0
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0answers
97 views

Engineering mathematics course and the order of the teaching of its topics

I am going to teach a course names "Engineering Mathematics" and the topics in it, are: Fourier series and integrals; including the motivation and computational aspects and their ...
51
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7answers
8k views

When is one 'ready' to make original contributions to mathematics?

This is quite a philosophical, soft question which can be moved if necessary. So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an ...
3
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1answer
126 views

Reference for “the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner”

I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded ...
92
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15answers
40k views

Top specialized journals

In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA. What journals ...
50
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11answers
10k 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 ...
4
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0answers
237 views

Are there topics of contemporary mathematics that simply require too much knowledge to be given as 3-year PhD problems? [closed]

Just out of curiosity, is there a topic where even an excellent graduate student wouldn't be able to master all the necessary background within 3 years (I say 3 because in many European countries a ...
5
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1answer
265 views

loop space of a finite CW-complex

Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...

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