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.
2,236
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What are the current breakthroughs of Geometric Complexity Theory?
I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods.
That program seems ...
10
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1
answer
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Alexandrov angles in Riemannian manifolds
Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...
395
votes
23
answers
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Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
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2
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Question on academic affiliation when submitting a paper
I just finished grad school, earning a Ph.D. in mathematics. Currently I do not know if I want to continue my academic career or not, but in the meantime I have written an article containing some of ...
6
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2
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Adding something to a book from an unpublished paper
As many of the people that I am spamming in real life might at this point know, I am turning my coend note into a book.
I would like to add a few pages taken from a (still) unpublished paper of mine (...
3
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0
answers
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Uses of excluded middle on a conjecture that can be rewritten constructively with this trick
An interesting proof technique is to use the law of excluded middle on a conjecture. There are proofs using LEM on the Riemann hypothesis for example.
Constructively this is disallowed (if you can ...
97
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11
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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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How should the Math Subject Classification (MSC) be revised or improved?
Most of us are familiar with the Math Subject
Classification
(MSC),
a coded index attempting to classify all mathematical
research areas by topic. The MSC, devloped jointly by the Math Reviews and ...
23
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5
answers
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What phenomena are better modelled by SDE instead of ODE?
Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
56
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10
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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 ...
10
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2
answers
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Is an easy proof of an interesting result worth publishing?
I am a Phd math student, and I was wondering what was the general consensus on what is publishable and what is not.
I've come here with an example in mind which should give some insight. I recently ...
13
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5
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Beginners text on calculus of variations
I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options.
I work on Machine Learning, and that where I intend to apply this.
...
25
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5
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What is some current research going on in foundations about?
What is some current research going on in the foundations of mathematics about?
Are the foundations of mathematics still a research area, or is everything solved? When I think about foundations I'm ...
14
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1
answer
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How exactly are realizability and the Curry-Howard correspondence related?
Consider, on the one hand:
the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...
123
votes
14
answers
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What are some noteworthy "mic-drop" moments in math?
Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in ...
8
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1
answer
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Examples of ZBMath reviews that motivated you to read the paper
This is community wiki question.
I will be writing my first review for ZBMath. I would like to take some suggestion through examples.
In general, abstract is too small and introduction is too lengthy ...
28
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8
answers
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Mathematical journals that accept long papers (up to 100 pages)
I have a problem with finding a suitable mathematical journal (maybe not of the highest level) that accepts longer paper (say 100 pages). I know only two such journals: Memoirs of AMS and Dissert.Math....
154
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54
answers
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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 use ...
102
votes
12
answers
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What is entropy, really?
I first saw the term "entropy" in a chemistry course while studying thermodynamics.
During my graduate studies I encountered the term in many different areas of mathematics.
Can anyone explain why ...
17
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6
answers
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Revisiting the unreasonable effectiveness of mathematics
Question:
On balance, with theoretical advances in algorithmic information theory and Quantum Computation it appears that the remarkable effectiveness of mathematics in the natural sciences is quite ...
2
votes
0
answers
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What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?
In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.
However he did not give any definition of $\mathcal{U}_\...
34
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4
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When has the scaffolding been more important than the completed building?
Niels Abel once said(1) of Gauss, "He is like the fox, who effaces his tracks in the sand with his tail." to which Gauss replied, "No self-respecting architect leaves the scaffolding in ...
69
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24
answers
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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 ...
43
votes
11
answers
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Open questions in Riemannian geometry
What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
3
votes
1
answer
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Sparse representation for continuous function?
I recently came across the field of "Sparse representation".
A talk is given here : https://www.youtube.com/watch?v=2bW4TkfTk-M.
The goal of sparse representation is taking a signal and ...
106
votes
15
answers
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Most striking applications of category theory?
What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
Joyal's ...
34
votes
6
answers
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Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
14
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2
answers
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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 ...
30
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3
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How much should the average mathematician know about foundations?
How much should an average mathematician not working in an area like logic, set theory, or foundations know about the foundations of mathematics?
The thread Why should we believe in the axiom of ...
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7
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How have mathematicians been raised? [closed]
Many of us have -- or at some point want to have -- children, and wonder how we can do our best to fulfill the "nurture" component of helping them develop mathematical talent... not because we want ...
68
votes
9
answers
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What are some important but still unsolved problems in mathematical logic?
In the past, first-order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...
6
votes
1
answer
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Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
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What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
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votes
91
answers
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Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
13
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1
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Mathematical fictionalism
Have there been any successful mathematicians that also happen to be mathematical fictionalists? Let's say success is defined by at least one article published in a non-pay journal.
I ask because ...
13
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3
answers
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What is so 'coloured' on Chromatic Homotopy Theory
As the title suggest, I would like know the motivation/ historical background
why chromatic homotopy theory is called 'chromatic'. Literally, what
analogy to colors it might have.
Accordings to
...
177
votes
80
answers
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Best online mathematics videos?
I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.
22
votes
2
answers
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Videos of Gian-Carlo Rota lectures
I apologize if this is off topic.
I think most of his listeners would agree with me that Gian-Carlo Rota had a wonderful style of lecture delivery. I have heard him lecture, both as an undergraduate ...
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8
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Computationally challenging integer sequences
I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good ...
280
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47
answers
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Examples of unexpected mathematical images
I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
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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 ...
5
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2
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How to effectively search Internet for graphs not for function graphs? [closed]
So, is there any way to distinguish graphs and plots in Internet?
I was looking for (Olivier-)Ricci curvature of graphs and found a lot about Ricci curvature of function graphs and not so much about ...
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3
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Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
91
votes
11
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What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
230
votes
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Your favorite surprising connections in mathematics
There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
38
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On critical reviews of Hawking's lecture "Gödel and the end of the universe"
The search for a neat Theory of Everything (ToE) which unifies the entire set of fundamental forces of the universe (as well as the rules which govern dark energy, dark matter and anti-matter realms) ...
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Does Physics need non-analytic smooth functions?
Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), ...
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answer
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How to select suitable journals for submission? [closed]
I think this post should be made community wiki, as there may not be some definitive answer, and it's more like a discussion.
My question is actually more subtle than the title of the post. What I ...
13
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1
answer
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Hausdorff and Naive Set Theory
Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following:
"Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as ...
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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 ...