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
Filter by
Sorted by
Tagged with
92 votes
0 answers
16k views

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 ...
Henry.L's user avatar
  • 7,951
60 votes
1 answer
6k 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 ...
30 votes
0 answers
3k 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 ...
Jim Humphreys's user avatar
29 votes
0 answers
3k 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 ...
25 votes
0 answers
588 views

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 ...
Nimas's user avatar
  • 1,267
24 votes
0 answers
847 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 votes
0 answers
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?
Vidit Nanda's user avatar
  • 15.4k
20 votes
0 answers
840 views

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 ...
Laura's user avatar
  • 343
19 votes
0 answers
882 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 ...
Asaf Karagila's user avatar
  • 38.1k
17 votes
0 answers
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 ...
Babai's user avatar
  • 280
16 votes
0 answers
412 views

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 ...
Modern_Hunter's user avatar
16 votes
0 answers
514 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 ...
user avatar
16 votes
0 answers
7k 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 ...
Moishe Kohan's user avatar
  • 9,664
13 votes
0 answers
355 views

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 ...
user333306's user avatar
13 votes
0 answers
787 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 ...
12 votes
0 answers
446 views

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 ...
cs89's user avatar
  • 971
12 votes
0 answers
1k 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 ...
user131781's user avatar
  • 2,442
12 votes
0 answers
692 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 ...
Zhen Lin's user avatar
  • 14.9k
12 votes
0 answers
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. ...
Michael Albanese's user avatar
12 votes
0 answers
806 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 ...
Tyler Holden's user avatar
12 votes
0 answers
268 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 ...
David Lewis's user avatar
10 votes
0 answers
337 views

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 ...
Max Muller's user avatar
  • 4,485
10 votes
0 answers
779 views

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 ...
JustLikeNumberTheory's user avatar
10 votes
0 answers
943 views

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 ...
xuq01's user avatar
  • 1,054
10 votes
0 answers
263 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 ...
Jules Lamers's user avatar
  • 1,813
10 votes
0 answers
836 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 ...
Stan's user avatar
  • 119
10 votes
0 answers
332 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 ...
sphere's user avatar
  • 413
9 votes
0 answers
1k views

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 ...
pie's user avatar
  • 139
9 votes
0 answers
550 views

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 ...
manifold's user avatar
  • 299
9 votes
0 answers
442 views

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, ...
Student's user avatar
  • 5,008
9 votes
0 answers
287 views

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}{\...
Asaf Shachar's user avatar
  • 6,611
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 ...
Tim Campion's user avatar
  • 60.6k
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 ...
user637140's user avatar
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 ...
user avatar
9 votes
0 answers
1k 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 ...
Francesco Polizzi's user avatar
9 votes
0 answers
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 votes
0 answers
278 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? ...
Jakob's user avatar
  • 1,986
8 votes
0 answers
169 views

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 ...
Dmytro Taranovsky's user avatar
8 votes
0 answers
171 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 \...
goblin GONE's user avatar
  • 3,693
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 ...
John Jiang's user avatar
  • 4,354
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 ...
Christopher King's user avatar
8 votes
0 answers
169 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 ...
HeinrichD's user avatar
  • 5,402
8 votes
0 answers
1k views

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 ...
Mikhail Katz's user avatar
  • 15.1k
8 votes
0 answers
397 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 ...
Noah Snyder's user avatar
  • 27.8k
8 votes
0 answers
583 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{...
SJY's user avatar
  • 579
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 ...
Xander Flood's user avatar
8 votes
0 answers
265 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 (...
RSG's user avatar
  • 421
8 votes
0 answers
325 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 (...
Mikhail Bondarko's user avatar
8 votes
0 answers
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 ...
7 votes
0 answers
300 views

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 ...
coudy's user avatar
  • 18.5k

1
2 3 4 5 6