Questions that ask about some aspect of mathematical research or study which doesn't involve the actual mathematics. In general, soft questions can be answered without using mathematical reasoning.

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3
votes
1answer
355 views

What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
22
votes
3answers
730 views

Graduate program applications that require questionnaires and other non-letter material

In the December 2014 AMS Notices, a letter to the editor (http://www.ams.org/notices/201411/rnoti-p1311.pdf) by Deconinck and Medlock addresses the problem of (math) graduate programs requiring letter ...
-1
votes
0answers
125 views

Introductory text to mathematics using type theory [closed]

[Beware: I'm using the nouns/concepts here quite loosely because I am searching for an introductory book on this issue.] As it turns out a set is an umbrella term for (at least) two concepts: A set ...
3
votes
2answers
456 views

Is a particular type of question about certain infinite sets still being asked?

I apologize in advance if this question is thought to be too soft or otherwise inappropriate for mathoverflow.net. Let M be the infinite set of all homeomorphism types of finite dimensional ...
10
votes
1answer
280 views

What are the current views on consistency of Reinhardt cardinals without AC?

It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...
3
votes
1answer
178 views

Decidability of Frankl's union-closed sets conjecture

Is it conceivable that Frankl's union closed sets conjecture is undecidable in $\mathsf{ZFC}$, or is this quite implausible, perhaps due to the "finitistic" nature of the statement, or for some other ...
3
votes
0answers
322 views

Well-known or prolific mathematicians that have never written a sole-author article? [closed]

Dear mathoverflow community, I have a junior colleague who will be coming up for tenure and who has written many articles, but all of them as a co-author. I don't see this as a problem (after all, ...
4
votes
1answer
140 views

What can be said about graphs if there are homomorphisms in both directions?

Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$. An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs ...
7
votes
3answers
1k views

The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...
24
votes
6answers
2k views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
2
votes
1answer
147 views

Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
4
votes
0answers
186 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 ...
16
votes
1answer
719 views

What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$. The problem seems to generate both proofs and disproofs at a fairly high rate, ...
0
votes
0answers
84 views

Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
0
votes
0answers
91 views

Proof of Faà di Bruno's formula with convolution identity

Faa di bruno's formula can be expressed as follows $$ \frac{d^n}{dx^n}[f(g(x))] = \sum_{k=1}^{n}f^{(k)}(g(x)) B_{n,k}(g'(x),....,g^{(n-k+1)}(x)) $$ On this Wikipedia page, there is a convolution ...
25
votes
6answers
2k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
8
votes
0answers
337 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. ...
16
votes
16answers
2k views

Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...
12
votes
1answer
722 views

Any reason I should join ResearchGate? [closed]

I am getting "invitations" to join ResearchGate. I am not a member of any other social network, as I consider it a waste of time. Are there good reasons for a mathematician to join ResearchGate? Can ...
37
votes
5answers
3k views

Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
12
votes
2answers
446 views

Who first noticed that Stirling numbers of the second kind count partitions?

When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity $$ x^n = ...
4
votes
0answers
147 views

Is the ISC kaput [closed]

The very useful Inverse Symbolic Calculator is showing me this What's up? multiple choice (a) No, it's fine at that address: idiot Edgar did something wrong... (b) It is off-line at that ...
2
votes
1answer
263 views

Disruptive innovations in mathematical notations [closed]

I am wondering whether there are examples of mathematical notations that, once introduced, have drastically changed or simplified the way to address a problem or a mathematical area, or that have ...
4
votes
4answers
282 views

Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?

Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is ...
12
votes
12answers
2k views

Obscure Names in Mathematics [closed]

I recently stumbled over the "Happy Ending Problem" (cf e.g. http://en.wikipedia.org/wiki/Happy_Ending_problem), which made we wonder, if there are other conjectures, theorems or problems, whose names ...
3
votes
2answers
424 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]

In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$ However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
2
votes
3answers
367 views

How did the summation operation come into use? [closed]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...
8
votes
0answers
420 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 ...
-1
votes
1answer
85 views

CAT spaces and Metric Measure Spaces [closed]

This is very vague question but here it goes: are there any results of analysis on metric spaces in CAT spaces?, for instance as in the paper of Sturm (On the geometry of metric measure spaces) or as ...
1
vote
0answers
120 views

current status of combinatorial optimization solvers [closed]

What is the current status of the solvers in combinatorial optimization? For example, what is the "usual feasible size" for a traveller sale's man problem, say, how many nodes and edges are "usually ...
1
vote
1answer
223 views

Concise definition of subobjects

Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, ...
2
votes
1answer
87 views

Interpretation of riemannian geodesics in probability

Good morning everybody. My question is, as maybe already hinted in the title, rather philosopic. We know that geometric properties of a riemannian manifold can be interpreted in terms of certain ...
10
votes
4answers
735 views

Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures ...
0
votes
0answers
174 views

Heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture

The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one ...
6
votes
0answers
184 views

Why is the Dynamical Mordell-Lang conjecture interesting?

The gist of the Dynamical Mordell-Lang conjecture is: Let's say you have a set $S$. Inside $S$, you have a point $x$ where you start off at. Now, you also have a function $f: S \rightarrow S$ (is ...
0
votes
0answers
48 views

Models for events where position and time are correlated

Apologies in advance if this question is not sufficiently research-level: What are the standard models that are used to describe phenomena in which events that occur at the same time are likely to be ...
2
votes
1answer
256 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
11
votes
3answers
1k views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
17
votes
8answers
6k 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 ...
5
votes
3answers
280 views

A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете I am surprised by the ...
64
votes
21answers
9k views

What is the most useful non-existing object of your field?

When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact ...
11
votes
2answers
818 views

What justification can you give for the fact that “most ODEs do not have an explicit solution”?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?
4
votes
1answer
175 views

Sites for seeking possible collaborations

As a material scientist, I have recently constructed algorithms for solving ground state of arbitrary cluster interactions models and prepared publications in the field of physics and material ...
6
votes
0answers
375 views

Any reason to believe that $NP \neq P$ is unprovable in ZFC

We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...
2
votes
0answers
323 views

Cases of Mathematical Fraud [closed]

Are there any cases of Mathematical Fraud? Analogous to other sorts of "scientific misconduct" it could be intentional [1], unintentional [2] or other [4]. [1]: Schoen Scandal ...
1
vote
0answers
93 views

reference on aperiodicity and cluster [closed]

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
2
votes
2answers
132 views

Software producing complex trees

Does anyone know any kind of graph software that could produce graphs like this for publication? Those links and crosses and numbers actually needs to be presented…. Thank you:) One small update, ...
2
votes
0answers
414 views

Help with my research topic [closed]

I have a masters degree in mathematics and I'm currently a PHD student. Since the beginning of my studies (2 years ago) I haven't progressed and still don't have a research topic. I was a very good ...
25
votes
9answers
1k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
15
votes
7answers
846 views

Where to find (personal) motivation [closed]

I think it would be appropriate to make this question CW... It is likely that this question will not survive here on MO for long, but I do hope that the community gives it a chance. I also hope to ...