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.
2,333 questions
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How do I learn to write mathematics professionally?
Hopefully this isn't too off topic.
I am an undergrad who really would like to one day do mathematics on a 'professional' level, but one hugely important area that I struggle in is communicating ...
7
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3
answers
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Relationship between algebraic number theory and analytic number theory
I’m curious about the relationship between algebraic number theory and analytic number theory from the following point of view:
Is it common for the two fields to have joint conferences?
Is it ...
0
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0
answers
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15 months with no feedback from Top 5 journal — Normal or time to withdraw? [migrated]
I am seeking advice regarding a submission to a Top 5 mathematics journal (analytic number theory).
Timeline:
September 2024: Article submitted, immediate acknowledgment from editor E1
March 2025 (6 ...
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0
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Recursive pointfree approach to algebraic topology
$\newcommand\seq[1]{\langle#1\rangle}$A large number of important topological results require simplicial-algebraic machinery (or comparable) to prove. This machinery is ingenious, impressively so even,...
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$p$-adic concepts having no analogue in the real case
I've always been fascinated how among completions of $\mathbb{Q}$, the real field $\mathbb{R}$, althrough historically more "ancient", seems to be the odd one out. Many interesting concepts ...
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0
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What happened to the second volume of Scharlau's biography of Grothendieck?
I am not sure whether this is the right place to ask this question, but I do not know a better place. Feel free to close it if you think that it is off-topic.
I recently discovered that the number ...
- 68.7k
4
votes
0
answers
250
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Simplicial complexes arising from Young diagrams and their homotopy type as wedges of spheres
My coauthors and I are finishing a paper in combinatorial and topological algebra, in which we define a class of simplicial complexes arising from Young diagramin the following way.
Let $\lambda = (\...
- 1,365
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2
answers
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Retiring early to pursue research in Pure Mathematics
I would like some advice in my situation. I am currently a medical student at the verge of graduation. I will start practicing after 2 years. Where I work, the pay is very good for physicians and I ...
19
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13
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Great theorems with elementary statements: 2026-onward [closed]
My 2021 book
Landscape of 21st Century Mathematics, Selected Advances, 2001–2020
collects great theorems with elementary statements published in 2001-2020. I now finishing the second edition of this ...
1
vote
1
answer
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Restrictions of metrics
A metric $d:M^2\rightarrow \mathbb{R}$ is any mapping that satisfies the well-known textbook definition of metric space $(M, d)$. In certain areas (optimal transport, topometric spaces, and quantum ...
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2
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How different are the categories $\mathbf{Ring}$ and $\mathbf{Ring} \hookrightarrow \mathbf{Rng}$?
When we define a group homomorphism $\theta \colon G \to H$, we do not have to specify that $\theta(e_G) = e_H$. On the other hand, most literature defines a ring homomorphism $h \colon R \to S$ with ...
- 241
6
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1
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Which exact sections of Higher Topos Theory are required for “Higher Algebra”?
I am currently reading Lurie’s “Higher Topos Theory” as a prerequisite for his book “Higher Algebra”. At the start of HA he mentiones that an exposition to $\infty$-categories is required and cites ...
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3
answers
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How to refer to Russian mathematicians?
I need to cite several Russian mathematicians in a paper and I would like some advice on how to write their names and how to write the bibliography. My main concern is to respect the name they were ...
- 307
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votes
1
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Known examples of conjectures stated while suspected false, to invite counterexamples?
Sometimes, in computational or experimental mathematics, one faces statements that seem almost certainly false yet are not directly refutable by current methods or feasible computation.
In such cases, ...
- 2,123
2
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4
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Which fields use homological algebra extensively?
I am currently very invested in homological algebra and since it is not a good research field itself (correct me if I’m wrong), I was wondering which fields use it much. My professor suggested ...
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4
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My dissertation is wrong, but I already defended. How to remedy?
When I first started working with my PhD advisor, he gave me a problem to work on. When my 5 years was about to be up, I had not published any papers but managed to write up solutions to two ...
17
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3
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Why it takes so long to referee a paper in some journals?
Some of the top mathematics journals typically take around two years to referee a paper, and sometimes even longer.
I am aware of a case in which a paper was under review for three years before being ...
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4
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Examples of mathematically interesting numbers that can be expressed as an integer tetration (besides Graham’s number)
It is well-known that Graham's number, $G$, can be expressed in radix-$10$ as a (very large) base-$3$ tetration with hyperexponent $n_{G} \in \mathbb{N}$ (i.e., $G := {^{n_{G}}{3}}$).
So, my natural ...
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5
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How can referees verify computationally intensive results when HPC resources are required?
This question is somewhat related to my previous question and is also inspired from this other question concerning the credibility of extensive computations (although from a different perspective).
In ...
- 1,365
6
votes
2
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Stone–Čech remainder of countable sets
In general topology, the most common example for countable space is $\omega$ with the discrete topology. Also, we "know" very well the topological properties and the structure of $\beta \...
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1
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697
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Source of a quote on boxed formulas
I vividly remember once reading (or hearing?) a claim that the level of a mathematical text is inversely proportional to the
$$
\boxed{\text{density of boxed formulas}}.
$$
Now I can’t find/remember ...
- 32.4k
12
votes
1
answer
955
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How to share algorithms for testing a conjecture?
I am preparing a paper where some results involve computational verification of a conjecture. Of course, I am not proving the conjecture in full, but I verify it for some large values of the involved ...
- 1,365
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10
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Which pairs of mutually contradicting conjectures are there?
Years ago I had the pleasure of witnessing Simon Thomas giving a wonderful talk about Martin's conjecture, which I just now fondly remembered reading this question. Even though I am not well-versed in ...
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6
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Graphical software tools for quick and easy diagrams
What tools do people use for quickly and easily creating presentable, if not publication quality, diagrams of various kinds?
When I need to make a high quality diagram, I'm happy to whip up some TikZ. ...
- 1,860
55
votes
9
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Improving readability of proofs
What do you do to improve the readability of finished proofs?
I basically found out that I keep a small mental checklist of criteria that I always go through after a proof is finished to improve the ...
56
votes
1
answer
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When did the OEIS get even better?
I'm asking this question out of curiosity, but also (and more importantly) to publicize to the research community something great that OEIS.org is doing.
Recently, I put a sequence into OEIS and got ...
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4
answers
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How long are you allowing yourself to be stuck on a problem? How do you know when to stop?
I searched for this question on the site but couldn't find it, so I'm asking it.
As a researcher, how long do you allow yourself to be stuck on a problem before deciding to move on? And how do you ...
61
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13
answers
4k
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How might mathematics have been different?
I think most people believe that mathematical truths are logically necessary. The fact that $\sqrt{2}$ is irrational doesn't depend on who proved it, when they proved it, whether they liked it, or ...
3
votes
0
answers
105
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Useful applications of convergence algebras
This is a soft question motivated by reading of "Convergence Structures and Applications to Functional Analysis" by Beattie and Butzmann.
A convergence algebra is a generalization of a ...
0
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1
answer
442
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Rook polynomial in Algebraic Geometry
The rook polynomial of a polyomino $\mathcal{P}$ is
$$
r_\mathcal{P}(t) = \sum_{k=0}^{r} r_k(\mathcal{P})\ t^k,
$$
where:
$r_k(\mathcal{P})$ is the number of ways to place $k$ non-attacking rooks on $...
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3
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Are there interesting finite groups which are not small?
I was playing around with finite groups recently, and a thesis (not really a conjecture, since it's rather informal) came to my mind that "all interesting behaviour of finite groups happens ...
- 1,774
3
votes
0
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372
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When did it become prevalent to only capitalize the first word (and names) in titles of journal articles? [closed]
Recently, I have noticed that it seems to be a relatively recent trend to only capitalize the first word (and names) in the title of a journal article. Some examples from the 21st century include:
...
- 30.7k
5
votes
1
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460
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When does the Kodaira symbol determine the Tamagawa number?
Let $E/K$ be an elliptic curve over a local field. I understand that the Kodaira type of $E/K$ refers to the isomorphism class of the special fiber of the Néron minimal model of $E/K$ as a scheme over ...
- 95
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1
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Compact function representation
One of my interest is 3D shape analysis, and there's a relatively recent framework called functional maps. In the framework of functional maps for shape analysis we represent shape features as a ...
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9
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Naming categories beyond their objects
Can you provide a known instance where it becomes necessary or useful to introduce a different name for the objects of a category and for the category itself?
Specifically, I am interested in cases ...
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5
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Note vs Notice in Mathematics [closed]
In common language there seems to be a difference between note and notice. However, I am discussing it with a co-author now and we are not sure about the usage in math. My feeling is that 'note' is ...
5
votes
2
answers
518
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Rigorous general treatment of degrees of freedom
My question is the following.
Is there an accepted mathematically rigorous and general treatment of the notion of "degrees of freedom" which at least accounts for its pervasive usages in ...
- 1,860
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1
answer
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How might fundamental mathematics differ for entities with intuitive comprehension of the continuum?
Dear MathOverflow Community,
I'd like to pose a speculative question, with apologies for its "soft" nature. My curiosity lies in how the day-to-day practice and the challenging frontiers of ...
- 489
2
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0
answers
142
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Partitioning a cube into cuboids of different dimensions
This is how I have tried:
Initial stage: One triplet of the form $(n,n,n)$.
Second stage: Decompose original triplet into two triplets by splitting one of the elements of $(x,y,z)$ into two parts at ...
- 221
1
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1
answer
530
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Why not begin studying sets with Hierarchy Theory?
By $\sf HT^\psi$ I mean the Hierarchy Theory of $\psi$ height. This is a set theory written in mono-sorted first order logic with equality and membership, with the following axioms:
Specification: $\...
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2
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0
answers
146
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Name for integral representation of Riesz potential
Let $\phi:\mathbb{R}^d \rightarrow \mathbb{R}$ be a sufficiently nice (e.g. Schwartz) radial function . Then it is classical by scaling that the Riesz potential $|x|^{-s}$, for $s>0$, may be ...
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Is this a real paper, and what to do with it [closed]
This paper appeared today on my arXiv digest (despite being submitted on April 3rd...): https://arxiv.org/abs/2504.21004v1. I started reading it because the title intrigued me, but I soon felt ...
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5
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How does the mathematical community handle minor, non-critical errors in published papers?
I’ve noticed that even peer-reviewed mathematical articles sometimes have minor errors, like small typos or slight logical gaps, which don’t affect the main results. I’m curious about how these kinds ...
3
votes
1
answer
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Leading journals in combinatorial commutative algebra
I am relatively new to research, and I am working in combinatorial commutative algebra. One of the things I find difficult is understanding which journals are considered strong or standard in this ...
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6
votes
1
answer
601
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How does collaboration work in math, especially differential geometry, research? [closed]
I understand how a math graduate program works; your supervisor is likely your first collaborator. Their collaborators or your fellow graduate students might also work with you on some papers beyond ...
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1
answer
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Are incidence algebras important?
I feel the need to explain my background before diving into this soft question, for you to understand my position.
During my undergraduate years, a Theoretical Computer Science professor asked me to ...
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3
answers
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What to do if paper was rejected with good report? [closed]
I submitted a paper to a good journal X. After four months it was rejected with no report. Nevertheless I asked if the editors have one. It turned out they did have a report which was rather positive. ...
0
votes
0
answers
98
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Is there a notion of approximate homotopies (similarly approximate homeomorphisms etc.?)
There are similar relaxations in other subjects. For example, there is a definition for approximate subgroups (see, e.g. here).
I was wondering if there are such constructs in topology too? One can ...
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Analysis of finite fields by further properties of the primes representing their characteristics
We know that finite fields have prime characteristic and we know a lot about them based in this fact. We can use that knowledge to establish very interesing and deep properties about these fields. In ...
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20
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Important unpublished works in mathematics
I know Harrington is quite famous for unpublished works in computability theory and a friend of mine specializing in set theory says Woodin is also quite similar which led me to the following question:...