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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77 views

Partitioning integers into two parts and exploring relationships with positional numeral systems

I asked this question in Mathematics StackExchange (link) about a month ago, but I have received no answer. It is about the following problem: Problem: Are there sets $A,B$ of integers such that $A\...
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1answer
283 views

Convergent and divergent thinking in a mathematical practice [closed]

Convergent and divergent thinking are two different approaches that are used in problem solving. Roughly speaking, convergent thinking is a focused search, attempting to find the single best ...
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0answers
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Specific versus general in titles of papers [closed]

It may not be possible to use mathoverflow in the way I'm hoping to, but my question is this: Should the title of a research paper in mathematics be as accurate and specific as possible (at the cost ...
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Translate a construction into categorical languages

Let $\mathscr C$ be a category whose objects are sets. Let $\mathscr V$ be another category. It seems that the usual composition in $\mathscr V$ can be somehow 'twisted' by $\mathscr C$. I notice that ...
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4answers
805 views

Examples of Mathematicians who excelled in Pure and Applied Mathematics [closed]

The other day I was thinking about mathematicians in history who made fundamental contributions to both pure and applied mathematics. The examples I can think of are Newton, Gauss, Euler, Archimedes ...
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7answers
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Decision problems for which it is unknown whether they are decidable

In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
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2answers
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What is the meaning of Text inside of AMS logo [closed]

What is the meaning of the text inside this AMS logo? The image is from here, and the logo seems to have been frequently used until the 80's. The text is ΑΓΕΩΜΕΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ but Google ...
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3answers
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Explanatory vs Non-explanatory Proofs [closed]

In a philosophical context, I’m currently thinking about how best to explicate mathematicians’ judgements that some correct proofs are ‘explanatory’ while others are not. In this vein, I’m trying to ...
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4answers
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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 ...
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1answer
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Ethics questions concerning a referee assignment

I recently refereed a paper that I returned to the author(s) for revision. The thrust of their argument relied on a claim whose justification I felt was lacking. I dutifully raised the issue in my ...
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0answers
186 views

Hodge decomposition and Kunneth formula on product manifold

Let $M$ be a compact oriented Riemannian manifold. Then we have the famous Hodge decomposition theorem: $$ \Omega^*(M)= im(d)\oplus \mathcal H^*(M) \oplus im(d^*) $$ Now, we want to consider the ...
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How much of concrete mathematics can be expressed in the language of category theory?

Question 1 How much of group/ring/lattice/... theory can be expressed in purely categorical terms (only using the notions object, morphism, identity morphism, and composition), that is, as properties ...
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Some questions in a paper by E. H. Neville (1949) about Farey series?

I am reading the paper MR0029924: Neville, E. H. The structure of Farey series. Proc. London Math. Soc. (2) 51, (1949). 132–144. (Reviewer: W. H. Simons) and by now two questions raised for me; ...
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Is there a research direction within dynamical systems theory / ergodic theory that concerns conjugability to a two-point motion?

Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if ...
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1answer
777 views

How can I improve my mathematical creativity? [closed]

NOTE: This post has been completely rewritten, but the ideas remain the same. I've been trying to figure out the divide between "good" and "great" mathematicians, and one metric I see repeatedly is "...
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0answers
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Modules over quasiisomorphic DG algebras

Suppose there is a quasiisomorphism $q: A \to B$ between DG algebras. Is there some reasonable description of induced functor $q^*: B-mod \to A-mod$? Can we say something better if it was a $B$-...
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4answers
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Some interesting and elementary topics with connections to the representation theory?

I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (...
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1answer
128 views

Meaning of “quantitative result” [closed]

Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
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3answers
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Applications of schemes to mathematical physics

Could anyone cite some applications or developments in mathematical physics or string theory that use schemes? I find curious the fact that while things like derived algebraic geometry and stacks ...
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3answers
689 views

Meaning of A-infinity relations

I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations. In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ ...
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6answers
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The work of mathematicians outside their professional environment

As it is reasonable to think the work of mathematicians will be developed/made in their offices of universities (or in eventual seminars or conferences), here are the colleagues, books and journals, ...
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15answers
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Examples of “unsuccessful” theories with afterlives

I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
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7answers
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Applications of mathematics in clinical setting

What are some examples of successful mathematical attempts in clinical setting, specifically at the patient-disease-drug level? To clarify, by patient-disease-drug level, I mean the mathematical work ...
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0answers
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How to be able to understand Cédric Villani's work? [closed]

I'm a grad student and I find Villani's work profoundly interesting, though I'm not able to read his papers. My question is, how can i cover the prerequisites to read them?
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0answers
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Gluing for derived category of coherent sheaves

Let $X$ be a scheme and assume $X=U \cup V$ for two affine schemes $U_0$ and $U_1$. If $\mathcal F'$ and $\mathcal F''$ are some (coherent) sheaves on $U$ and $V$ respectively such that $\mathcal F'|_{...
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2answers
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How to accelerate progress in mathematical research? [closed]

After completing a Ph.D. in pure mathematics, 10 years ago I left academia for working in industry. There, a typical question is "What can we do to accelerate $x$?" when a project is slowed down, and ...
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1answer
593 views

Highly flexible PhD programs/school (with individual freedom)

Do you know of any PhD programs that are highly flexible? I'm interested in either math or computer science programs, as I have done research in both areas and feel confident to apply to both. With ...
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5answers
609 views

Famous conjectures named after a mathematician that were resolved in their lifetimes [closed]

This is a question that I thought about recently, and I thought would be interesting to the MO community. What are some famous conjectures, more specifically those that attracted a lot of attention ...
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0answers
573 views

Next step in studying arithmetic geometry

This relates to this post. I want to study arithmetic, such as Fermat's last theorem, Faltings' theorem, Mazur's torsion points theorem, Weil conjecture and so on. For understanding these theorems (...
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0answers
668 views

Have any of Maryam Mirzakhani's doodles been preserved?

I edit a magazine for High School students, and would very much like to get hold of a large image of one of the large sheets of paper with Maryam Mirzakhani's mathematical drawings on for the cover of ...
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6answers
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Papers on arXiv solving the same problem at the same time

I am little bit curious about the following examples at least two papers appeared on arxiv at the same day solving one and the same problem. Have you ever seen such a coincidence? If yes, can ...
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1answer
442 views

Odd differential forms

In de Rham's classical book "Variétés Différentiables" de Rham, Georges, Variétés différentiables. Formes, courants, formes harmoniques. 3e éd. revue et augmentée, Publications de l’Institut de ...
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1answer
960 views

Which mathematician sampled published proofs and found one third of them to have errors?

A recent question about whether/how we can trust mathematics in the face of human fallibility reminded me of a paper or article I read probably more than twenty years ago about a mathematician who was ...
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13answers
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Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?

To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...
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1answer
335 views

Survey article model theory research

I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model ...
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15answers
4k views

Unconventional examples of mathematical modelling

I'll soon be teaching a (basic) course on mathematical control theory. The first part of the course will focus on mathematical modelling of dynamical systems. More precisely, I will present examples ...
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1answer
1k views

Who invented Monoid?

I was trying to find (and failed) the original author of either the concept of Monoid (set with binary associative operation and identity) the name (which sounds french ? and also Dioid (for what ...
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3answers
4k views

On math looking obvious in retrospect [closed]

Admittedly, a soft-question. I, being a very young researcher (PhD student) have personally faced the following situation many times: You delve into a problem desperately. No progress for a very long ...
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4answers
1k views

Reference request: any 20th century German critiques of Bourbaki? [closed]

Vladimir Arnold is known, among other things, for offering a scathing critique of Bourbaki: The Arnold – Serre debate Recently I've been reading some Nietzsche, and he chides some Germans in the ...
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1answer
142 views

Graphs with Hermitian Unitary Edge Weights

Very recently, Hao Huang proved the Sensitivity Conjecture, which had been open for 30 years or so. Huang's proof is surprisingly short and easy. Here is Huang's preprint, a discussion on Scott ...
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2answers
1k views

Why would one number theorems, propositions and lemmas separately?

When it comes to numbering results in a mathematical publication, I'm aware of two methods: Joint numbering: Thm. 1, Prop. 2, Thm. 3, Lem. 4, etc. Separate numbering: Thm. 1, Prop. 1, Thm. 2, Lem. 1,...
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4answers
487 views

Texts on moduli of elliptic curves

I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves. I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...
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2answers
231 views

Naming convention: Adjective for linear operators that are endomorphisms

If a matrix has the same number of rows and columns, we call it a square matrix. The analogous concept for linear operators would be operators with the same domain and range, i.e., endomorphisms. Is ...
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1answer
212 views

Are there major research areas in math? Or is it a lot of individual efforts? [closed]

In physics, for example, dark matter is a major research area now. And there are specific parts of that trending. Is there anything similar in math? Is there something the majority of mathematicians ...
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0answers
350 views

Is there such a field as applied $\infty$-category theory?

It seems that applied category theory has exploded in popularity in recent years. My question is simple: had there been any work using $\infty$-category theory in applications? Edit: By ...
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27answers
5k views

Examples of simultaneous independent breakthroughs

I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. ...
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0answers
172 views

What imaginations of Lebesgue spaces or other Banach spaces do people intuitively share?

At several occasions I heared people discussing about the „colors“ of Lebesgue spaces $L^p$: $L^2$ is red, $L^1$ is white, $L^\infty$ is black, and the other $L^p$ are blue or violett. Of course this ...
14
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1answer
497 views

Legendary extra parameters to simplify a counting problem

I am reading Proofs and Confirmations, the history behind the alternating sign matrix conjecture, regarding counting $n \times n$ alternating sign matrices. In the introduction, it is written that ...
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1answer
227 views

Limits of a family of integrals

Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals. QUESTION. What is the value of this limit? It seems to exist. $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...
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5answers
8k views

Misspelling my name on my mathematical publications

Perpetuating a mistake by my thesis advisor, I misspelt my name on my mathematical publications so far -- any advice on what to do? As it happens, my thesis advisor, with whom I co-authored my first ...

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