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.
253
questions with no upvoted or accepted answers
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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 ...
60
votes
1
answer
6k
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Why "open immersion" rather than "open embedding"?
When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
30
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0
answers
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Greatly expanded new edition of a Bourbaki chapter on algebra?
Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
29
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3k
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The Work of Pierre Deligne
In this biography of Pierre Deligne, there is a quote of Jacques Tits which says that "quite a few of his best ideas have never been written!".
What are some of his best ideas that you have heard of ...
25
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0
answers
588
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Galois representations attached to Shimura varieties - after a decade
In an answer to the question Tools for the Langlands Program?, Emerton, in his usual illuminating manner, remarks on the reciprocity aspect of Langlands Program: "...As to constructing Galois ...
24
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0
answers
847
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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 ...
21
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3k
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Origins of the Nerve Theorem
Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?
20
votes
0
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840
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A mysterious paper of Stallings that was supposed to appear in the Annals
In Stallings's paper
Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389.
he briefly discusses how to prove "several generalizations" of Brown's ...
19
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0
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882
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What examples of existence forcing proofs are there?
Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing.
There are only a handful of ...
17
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0
answers
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Etale Slice Theorem
I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?
This is Luna's Slice theorem from a ...
16
votes
0
answers
412
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Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
I ...
16
votes
0
answers
514
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Reference request for Grothendieck's work on "Integration with values in a topological group"
Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
16
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0
answers
7k
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Story of "Grothendieck's prime number" 57
I asked this question earlier, at hsm.stackexchange.com without much luck. Maybe somebody can answer it here.
There is a story about Alexander Grothendieck and the "Grothendieck Prime" 57, which ...
13
votes
0
answers
355
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Context of set theory in which one doesn't have to worry about size issues
In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck:
It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories ...
13
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0
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787
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Mathematical etiquette: Rephrasing / restructuring a work, limited release (with attribution) acceptable?
I am reading a mathematics textbook (which one is irrelevant, and I do not wish to insult the author if (s)he happens to be reading this). One section relies quite a bit on an appendix and results ...
12
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0
answers
446
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Where to upload digitized LaTeX versions of old papers?
Let's say I took the time to convert into proper modern LaTeX a few old math papers (which cannot be found online, or only in poorly scanned versions which are hard to read due to this). If someone ...
12
votes
0
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1k
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The status of the journal “Forum Geometricorum”
The online journal Forum Geometricorum is a sort of central organ of elementary geometry (mainly triangle geometry and related topics). It has been published regularly since 2000 but seems to have ...
12
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0
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692
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"To operate the machine, it is not necessary to raise the bonnet."
The quotation in the title is attributed to Frank Adams and appears in several places:
In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not necessary ...
12
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0
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1k
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How much algebraic geometry do I need to study complex geometry?
As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...
12
votes
0
answers
806
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State of research in moduli space of flat connections
I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...
12
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0
answers
268
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Eilenberg's rational hiererchy of nonrational automata & languages — where is it now?
In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised Volumes C and D dealing with "a hierarchy (called the rational ...
10
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339
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Examples of games developed purposely to analyze players' strategies for mathematics research
Background
This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
10
votes
0
answers
780
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How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
10
votes
0
answers
945
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The "unification" of geometry via topos theory?
This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question.
There has been quite a lot ...
10
votes
0
answers
263
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Mirror site for the NIST Digital Library of Mathematical Functions (DLMF)
For research I use the NIST Digital Library of Mathematical Functions (https://dlmf.nist.gov/) quite often for looking up basic facts about special functions. At the moment, however, this gives
...
10
votes
0
answers
836
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Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
10
votes
0
answers
332
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Algebraic K-theory of a ring.
I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ?
I ...
9
votes
0
answers
1k
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How complicated can an elementary antiderivative get?
I asked this question on MSE here.
I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
9
votes
0
answers
550
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What is the status of N. Durov's PhD thesis?
N. Durov Phd thesis "New Approach to Arakelov Geometry" is ofted mentioned as a beautiful approach to Arakelov geometry and it includes also a treatment of $\mathbb F_1$. It is a very long ...
9
votes
0
answers
442
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Useful applications of applied category theory
Led by John Baez, applied category theory (e.g. [1]) seems to accumulate much popularity. As someone who has noticed the importance of category theory in pure mathematics (e.g. homotopy theory, tqfts, ...
9
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0
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287
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Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?
$\newcommand{\id}{\operatorname{Id}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\...
9
votes
0
answers
675
views
Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
9
votes
0
answers
493
views
How bad is it to publish a paper with an overcomplicated proof?
I don't want to go into details for anonymity purposes, but I have co-authored and submitted a paper with a long proof (dozens of pages), and I think that with some moderate effort, we could find a ...
9
votes
0
answers
626
views
étale vs syntomic vs flat cohomology
Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
9
votes
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answers
1k
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A quote by Lev Landau about prime numbers
I was talking with a student of mine about Goldbach's conjecture, and a certain point he asked why this apparently simple statement is so difficult to prove.
Half-joking, I answered "well, because ...
9
votes
0
answers
2k
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"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
9
votes
0
answers
278
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Tangent space, metrics etc. on simplicial sets
Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...
8
votes
0
answers
169
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Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
8
votes
0
answers
171
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Does the "coproduct-elimination transform" have an accepted name, and where can I learn more about it?
Suppose we're in a bicartesian closed category. Then given a morphism $$f : X \rightarrow Y_1 + \ldots + Y_n$$ and a test object $T$, we get a corresponding morphism
$$T^f : X \times [Y_1,T] \times \...
8
votes
0
answers
367
views
Online study groups for individual mathematical texts
This has been one of my earlier academic dreams. I have pitched the idea to Prof Ravi Vakil, among others. Recently, because of participation in the mathematical competition hosted by Alibaba, I have ...
8
votes
0
answers
650
views
Is there any theorem achieving Conway's "Mathematician's Liberation Movement"
John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
8
votes
0
answers
169
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Sharply less regular cardinals in set theory
If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible ...
8
votes
0
answers
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What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
8
votes
0
answers
397
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Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
8
votes
0
answers
583
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Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?
This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are
similarities between $\mathbb{Z}$ and $\mathbb{...
8
votes
0
answers
418
views
History of the characterization of commutative Artin rings
When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...
8
votes
0
answers
265
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Fixed marginals of joint distribution: status
One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left (...
8
votes
0
answers
325
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Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?
The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (...
8
votes
0
answers
1k
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triangulated/derived categories in Physics and algebraic geometry
Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...
7
votes
0
answers
300
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A citation from Dieudonné about existence, classification and construction
I am looking for an exact quote from Jean Dieudonné saying more or less that
there are three kinds of important problems in mathematics:
problems of existence, classification and construction.
The ...