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.

196 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 ...
48
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2answers
5k views

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 ...
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2k views

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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2k views

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 ...
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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 ...
21
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546 views

Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory ?)

The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...
19
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816 views

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 ...
19
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0answers
3k views

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?
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405 views

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 ...
15
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719 views

Application of higher categories in algebra

Higher topos and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categories ...
15
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0answers
1k views

Grothendieck's manuscripts

This is a curiosity question. Some time ago, late Grothendieck's manuscripts have been uploaded, with consent from Grothendieck's family, for the community to consult. They can be found here, ...
14
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0answers
555 views

Does inner model theory seek canonical models for large cardinals?

Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
14
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733 views

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 ...
13
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2k views

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 ...
12
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663 views

“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 ...
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1k views

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
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766 views

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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243 views

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 ...
11
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621 views

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 ...
10
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355 views

Recommendations on PhD programs in Europe on number theory

I am a Master student of math in the UK, and I would like to do a PhD in number theory, preferably the algebraic side of it. Given that I am in the UK, I would probably prefer staying here or going to ...
10
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2k views

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 ...
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292 views

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 ...
10
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660 views

Use of an appendix in a long paper

I am writing a long paper (around 100 pages). I would consider 50 pages of it interesting in that it solves a problem of some significance in my field and contains an number of difficult ideas in the ...
9
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554 views

Why is Planar algebras I (by Vaughan Jones) not published?

On Saturday 4 September 1999, Vaughan Jones put on arXiv a paper entitled Planar algebras, I. Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the ...
9
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300 views

Does anyone use non-sober topological spaces?

Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point. Is there any area of mathematics outside of general topology where non-...
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955 views

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
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736 views

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 ...
9
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793 views

Refereeing a paper containing strong statements about other papers

The title says it all. Suppose you are refereeing a paper where the author A makes strong statements about other papers by a different author B, like: the proof of Theorem 1 in paper [B] is wrong and ...
9
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2k views

“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
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229 views

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
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139 views

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 \...
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493 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 ...
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169 views

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 ...
8
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318 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 ...
8
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155 views

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
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527 views

Errata in EGA, collected

There is an extensive list of EGA's errata on the books themselves, but my question is whether new errata, that is those found by various mathematicians after the publication, are collected somewhere. ...
8
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361 views

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
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545 views

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
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0answers
367 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
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241 views

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
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303 views

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
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1k views

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 ...
8
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1answer
663 views

How many graduate students in mathematics are there worldwide?

There is already a question about the number of research mathematicians here, but I was wondering : is it possible to estimate the number of graduate students in mathematics worldwide in 2019 ? (...
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308 views

The meaning of this mysterious remark in Littlewood's Miscellany

In the well known book by Littlewood (Mathematician's Miscellany, or the later edition called Littlewood's Miscellany) there is a remark made in the chapter 'A Mathematical education', the meaning of ...
7
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113 views

A generalization of matrix minors to non-integer values

I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$ One approach I thought of was to use the fact that the $k$-minors are (...
7
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0answers
194 views

Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$

Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ? I have looked at the very few books on cohomology of groups ...
7
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409 views

What is the geometric intuition for the sheaf-theoretic terms “soft”, “fine”, and “flabby”?

The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here). ...
7
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0answers
390 views

Representation theory of symmetric group for dummies

I have to grade a master project on representations of symmetric groups (char $0$) third time in my life and finally I came to a conclusion that I have to get a grasp of the matter. I am aware of ...
7
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0answers
340 views

What Spec-like functors are there?

The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there ...
7
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0answers
353 views

What would you do if you improve your own result that is submitted but not publishied?

Here is a hypothetical situation: You have proved a result and written up a paper about it. You have submitted your article to some journal and it is being reviewed. While you are waiting, you have ...