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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14 votes
2 answers
1k views

$n!$ 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: ...
0 votes
0 answers
413 views

Solving the equation $\operatorname{Powerset}(X)=\varnothing$

There are (at least) two variants of this question. Is it possible to modify the axioms of set theory, without arriving at obvious contradiction, in such a way that in a model of the theory there is ...
8 votes
1 answer
464 views

In search of a combinatorial reasoning for a vanishing sum

Assume $s, j \in\mathbb{N}$. Define the set $$\mathcal{A}_{j,s}:=\{(n_1,n_2,\dots,n_j)\in\mathbb{Z}_{\geq0}^j\vert \, n_1+2n_2+\cdots+jn_j=j, \, n_1+n_2+\cdots+n_j=s\}.$$ Question. Is there a ...
7 votes
6 answers
2k views

Elegant representations of graphs in R^3

If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any ...
6 votes
0 answers
256 views

Cohomology theories from Saito's mixed Hodge complexes

The definition of mixed Hodge complexes by Saito is a very interesting one, since it's more a cohomology theoretic than geometric generalization of Hodge structures. Since Saito's motivation for mixed ...
8 votes
1 answer
393 views

Can ETCC/ETCS talk about 'size issues'?

In material set theories (like ZFC), one can prove that there is no set of all sets. Can one prove a similar statement in ETCS? This exact statement "there is no set x such that y in x for every set y"...
37 votes
5 answers
5k views

Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
1 vote
1 answer
235 views

partition theory: meet the COP

Recall that $(a;q)_0:=1,\,(a;q)_n=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ and $(a;q)_{\infty}=(1-a)(1-aq)(1-aq^2)\cdots$. Let's introduce the following (generalized) concept. A colored overpartition (...
3 votes
0 answers
78 views

A complex differential theorem applying to compact projective manifolds but not all compact Kahler manifolds

The following question may be soft, but I hope it is precise enough. The Hodge conjecture, if proven, would be a theorem in complex differential geometry that holds for all compact projective ...
8 votes
5 answers
3k views

Alternative for Kadison and Ringrose's book

I have read over the book by R. V. Kadison and J. R. Ringrose: Fundamentals of the theory of operator algebras. Vol 1, and have done most of the exercises in it. Now I want to find an alternative book ...
1 vote
0 answers
291 views

Finding Good Papers To Learn From [closed]

I was reading a question on here about the road to learning arithmetic geometry, and one of the suggestions was to start reading some foundational papers in the area. Similarly, one of the responses ...
29 votes
3 answers
3k views

Etiquette of publishing folklore results

I am wondering what is the etiquette of publishing a "folklore" result? Though special cases of the result are well-known, the proof is not readily available in any reference text or paper I've seen....
23 votes
1 answer
2k views

Etymology of "exterior" in "exterior calculus"

What is the origin of the term "exterior" in "exterior calculus"? How does this term relate to "interior products" and "inner products", if it does at all?
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 ...
11 votes
2 answers
3k views

Is it fine to inquire about a paper that's been under review for around 9 months?

I have submitted a paper on applied probability in one of SIAM journals. The paper is under review for 9 months. I asked the editor 1 month ago about it, I was told that one review report has come and ...
8 votes
2 answers
6k views

Equal signs with fancy marks

Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions. In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...
7 votes
1 answer
978 views

Does equality between sets contradict the philosophy behind structural set theory?

Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. This means that for every two objects/sets $a,b$ one can ask ...
8 votes
1 answer
2k views

What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes, Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor ...
8 votes
1 answer
1k views

Recursion theory from the standoint of category theory

It is (I believe) a very easy exercise to prove that the general recursive functions over the natural number object $N$ form a category. But what sort of category is it? From the fact that one can ...
1 vote
2 answers
2k views

Journal with good reputation on Dynamical Systems and Complex Dynamics [closed]

I am wondering which journal are considered good to publish at in the areas of Dynamical Systems (Hamiltonian Dynamics) and Complex Dynamics. I have a tendency to ignore completely the journal in ...
5 votes
1 answer
317 views

Convention on Clifford Product [duplicate]

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign: $vv=Q(v)$ (see, for instance, Wikipedia) $vv=-Q(v)$ (see, for instance, MathWorld or ...
2 votes
2 answers
702 views

Are simplified elementary proofs if valid interesting to the professional mathematical community [closed]

For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the history of Fermat's Last Theorem. I have made numerous mistakes ...
5 votes
1 answer
390 views

Rigorous introductions to actuarial mathematics

Maybe this is not the kind of question for this website, but nevertheless I believe it could be interesting for a large audience. I am interested to know if there is some book where the subject of ...
5 votes
1 answer
430 views

Using High Level Probability Theory (eg Markov Chain Mixing) in Cryptography/Cryptanalysis

I'm currently doing a PhD in probability theory, specifically (discrete space) Markov chains and their mixing properties. As well as my current main project, I'm looking to have a side project, eg to ...
47 votes
14 answers
21k views

Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
27 votes
8 answers
13k views

Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such. Yet, they could have been just ...
13 votes
7 answers
34k views

Real analysis has no applications?

I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
16 votes
5 answers
6k views

Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory. So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...
9 votes
3 answers
2k views

Need examples of homotopy orbit and fixed points

I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or ...
19 votes
2 answers
2k views

Which kind of foundation are mathematicians using when proving metatheorems?

Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. For every two objects/sets $a,b$ one can ask whether $a=b$ or not....
1 vote
0 answers
69 views

Soft: Lagrange Multiplier and Intersection of Thickened Sets

Suppose I have an optimization problem of the form $$ \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x), $$ for some convex function $f$ and non-convex l.s.c. function $g$. Can we reinterpret the ...
45 votes
14 answers
12k views

Examples of undergraduate mathematics separation from what mathematicians should know

I'm looking for examples of four kinds of things: Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying ...
88 votes
9 answers
14k views

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 ...
10 votes
3 answers
4k views

Looking for Arnol'd quote about Russian students vs western mathematicians

I think I once saw a sentence in an article by V.I. Arnol'd saying something like: here is a problem that every Russian schoolchild can solve, but no western mathematician can solve. But I can't find ...
10 votes
8 answers
4k views

Most important mathematical results in last 30 years [closed]

Which results from the last 30 years, in any area of mathematics, do you think are the most important ones? Specifically, which are the ones that will have more impact across all math and/or settle ...
26 votes
2 answers
8k views

Software for symbolic matrix calculus?

Is it possible to get widely available math software (Maple/Matlab/Mathematica, etc) to symbolically differentiate vector and scalar functions of matrices, returning the result in terms of the ...
3 votes
0 answers
272 views

Looking for author of calculus quote

When I was a lowly calculus student many many years ago, my calculus teacher quoted some famous mathemtician: "Calculus is the last course in arithmetic and the first course in mathematics that one ...
6 votes
2 answers
2k views

Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...
0 votes
1 answer
196 views

Analysing/studying simple groups with Sylow-$2$ subgroups

In the classification of finite simple groups, some classification is done by considering structure of Sylow-$2$ subgroups (for example, here; it is more than 250 page paper!) Now in the world of ...
5 votes
2 answers
334 views

Formal Definition of Finite Conditions

Forcing with finite conditions is a common concept used by set theorists. I was thinking about its meaning, but I couldn't find any exact definition of it. At the first glance it seemed to me that ...
7 votes
0 answers
587 views

How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]

Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...
2 votes
0 answers
193 views

Locally consistent theories

Is there a notion in the literature of a theory which is locally consistent, in some sense wherein choosing a "vantage" yields a theory which is consistent, but the entire theory may not be consistent?...
8 votes
1 answer
3k views

Hierarchy of Grothendieck's SGA, EGA, FGA

I was thinking about a possible hierarchy for the top three Grothendieck's works: EGA,SGA,FGA. But I haven't read all these works, and so I'm asking if there is actually such a hierarchy. Here the ...
271 votes
67 answers
136k views

Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
26 votes
4 answers
3k views

Would you resubmit a research paper after it has been superseded by another as yet unpublished paper?

Hope that the following soft question is still appropriate on MathOverflow. I was wondering if there is any communal protocol or etiquette with regard to the resubmission of a research paper after it ...
6 votes
1 answer
576 views

What structure do you get if you adjoint a root of $z \bar{z} = -1$ to the complex numbers?

A pop-up is informing me that my question is likely to be closed. Still, recall that the complex numbers $\mathbb{C}$ was conceived by trying to adjoint a root of the equation $x^2 = - 1$ to the ...
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 ...
6 votes
1 answer
414 views

Dyson's invitation: Opportunities in juxtaposition of incompatibles

"Up to now, my examples of missed opportunities have been mathematical discoveries which actually occurred, although they could have occurred a long time earlier. In such cases one can be sure that an ...
1 vote
0 answers
218 views

What should one know about abstract sets and structural foundations?

Recently I came by accident across the book sets for mathematics by Lawvere. It says: First we deplete the object of nearly all content. We could think of an idealized computer memory bank that ...
10 votes
2 answers
420 views

Why is the notion of algorithm a primitive one in Brouwer's intuitionism?

I've seen several times people mentioning that the notion of an algorithm / a computation is taken as a primitive notion in L. E. J. Brouwer's intuitionism. For instance, in Varieties of Constructive ...

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