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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Consequences of the Langlands program
In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture.
I would like to know: are there any ...
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$n!$ divides a product: Part I
Question. The following is always an integer. Is it not?
$$\frac{(2^n-1)(2^n-2)(2^n-4)(2^n-8)\cdots(2^n-2^{n-1})}{n!}.$$
John Shareshian has supplied a cute proof. I'm encouraged to ask:
...
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1
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How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
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Where to publish a long classification?
Suppose that the classification of some mathematical (say algebraic) notions requires (say) 70 pages. Let clarify that (say) 90% of the pages are used to write the result itself, whereas only 10% are ...
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'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math
To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?
To put it long: I know almost nothing about category theory - but ...
11
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1
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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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Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?
I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, ...
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Advice on choosing an area of specialization [closed]
I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...
8
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2
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Are trivial zeros of the zeta function important?
Non-trivial zeros play an important (main) role in the distribution of prime numbers.
Are there theorems in which trivial zeros play an important (main) role?
user111966
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Books and resources on PDEs that use Mathematica and Matlab
Can you recommend some reference books that use software like MATLAB and Mathematica to deal with the basic topics in
analysis of PDE (the ones you can find in Strauss' book Partial Differential ...
user60665
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2
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Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?
I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the ...
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2
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Common/well-known results with natural and/or useful reformulations
$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that
the reformulation/...
6
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3
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What is special in dimension $2$ (When characterizing isometries using the cofactor matrix)?
Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \...
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Terminology introduced in recent years with more than one meaning
Suppose a term(inology) is recently (in last 20 years) introduced in research mathematics.
It might happen that some one who wish to use it, in the same area of research, for different purposes or ...
4
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What solutions to useful computational problems could be rewarded through cryptocurrency smart contracts?
What kinds of cryptocurrency smart contracts could be used to reward people for solving specific kinds of useful computational problems?
Background
In this question, I asked for proposals for useful ...
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0
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Variational formulation for elliptic interface problem
Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
user60665
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0
answers
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Confusing notation for sets of unordered vs ordered pairs
Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...
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0
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Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
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1
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A topologist is not a mathematician - a small question
Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the ...
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Companion to theoretical physics for working mathematicians
In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
254
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41
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A single paper everyone should read? [closed]
Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.
Do ...
224
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How can a mathematician handle the pressure to discover something new?
Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...
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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 ...
183
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127
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Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
158
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Resources for mathematics advising.
This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has ...
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11
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Why are flat morphisms "flat?"
Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...
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Proofs shown to be wrong after formalization with proof assistant
Are there examples of originally widely accepted proofs that were later discovered to be wrong by attempting to formalize them using a proof assistant (e.g. Coq, Agda, Lean, Isabelle, HOL, Metamath, ...
148
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26
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Good "casual" advanced math books
I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...
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How do you keep your research notes organized? [closed]
One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the ...
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Pressure to defend the relevance of one's area of mathematics
I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “...
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38
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Noteworthy, but not so famous conjectures resolved recent years
Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch ...
114
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What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?
In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\...
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New grand projects in contemporary math
When I was a graduate student in math (mid-late eighties and early nineties) the arena was dominated by a few grand projects: for instance, Misha Gromov's hyperbolic groups, which spread into many ...
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Examples of notably long or difficult proofs that only improve upon existing results by a small amount
I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros ...
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Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
102
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8
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When should a result be made into a paper?
I recently posted a short (6 page) note on arXiv, and have more or less decided that I should not submit it to a journal. I could have tacked it onto the end of a previous paper, but I thought it ...
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How has "what every mathematician should know" changed?
So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are ...
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8
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Which are the best mathematics journals, and what are the differences between them? [closed]
Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to?
Coming up with a shortlist isn't ...
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Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
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Peer review 2.0
I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many ...
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5
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Is there a database for tracking the dependencies of mathematical theorems?
Given a proof for a result, one could denote the proof as a node on a graph, and then draw arrows to the node from axioms and previous results that the proof uses, and then draw arrows from the node ...
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What's a great christmas present for someone with a PhD in Mathematics?
Christmas is just around the corner and I haven't bought all the gifts for my family yet ( yeah, 😢)
My Dad has a PhD in Mathematics, he works in Graph theory and his thesis was about Quasiperiodic ...
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Hironaka's proof of resolution of singularities in positive characteristics
Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...
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Time-saving (technology) tricks for writing papers
I have over the years learned some tricks which saves a lot of time,
and I wish I had known them earlier. Some tricks are LaTeX-specific, but other tricks are more general. Let me start with a few ...
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6
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Why didn't Vladimir Arnold get the Fields Medal in 1974?
As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the ...
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Books on music theory intended for mathematicians
Some time ago I attended a colloquium given by Princeton music theorist Dmitri Tymoczko, where he gave a fascinating talk on the connection between music composition and certain geometric objects (as ...
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Modern Mathematical Achievements Accessible to Undergraduates
While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
88
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Work of plenary speakers at ICM 2014
The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM ...
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Where is number theory used in the rest of mathematics?
Where is number theory used in the rest of mathematics?
To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to ...
86
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5
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What is sheaf cohomology intuitively?
What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...
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