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. In other words, questions that can be answered without making computations or applying theorems and axioms.

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0
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1answer
108 views

Heat Equation on LCA Group

I am reading The Laplacian on A Riemannian Manifold. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation ...
9
votes
3answers
603 views

How to refer to plural of mathematical symbols - with or without an apostrophe [closed]

Which one is correct, $x_i$s or $x_i$'s? Example sentence: The $x_i$s form a sequence. The $x_i$'s form a sequence.
7
votes
1answer
688 views

Research and exposition: how does writing “basic” books affect your “serious” research work?

I can see the benefit of writing a mathematical monograph: you revise and organize your own work and recollect the key ideas of your own research. But this applies only to books aimed at researchers ...
49
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4answers
4k views

Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...
1
vote
0answers
227 views

On the remainder term in Taylor's formula [closed]

(1) What are the main differences, in terms of "usefulness" while solving problems (even at research level), among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula? Could ...
12
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2answers
1k views

New research and re-discovering classic results in “basic” real analysis

Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, ...
31
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7answers
3k views

Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
25
votes
5answers
2k views

Collaboration or acknowledgment?

This post is a sequel of: When should a supervisor be a co-author? This time the topic is about the interaction between two professional mathematicians (in particular junior-senior, but not ...
18
votes
5answers
1k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
6
votes
2answers
391 views

Generalization of Sylvester-Gallai theorem

The Sylvester-Gallai theorem states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single ...
5
votes
1answer
172 views

Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
6
votes
1answer
271 views

Is there a “good” reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?

Let $$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$ be the braid group on three strands, and consider the surjection $$\phi : Br_3 \twoheadrightarrow ...
23
votes
3answers
906 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 ...
3
votes
2answers
476 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 ...
11
votes
1answer
326 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 ...
4
votes
1answer
226 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
353 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
156 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 ...
22
votes
7answers
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
166 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
223 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
747 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
98 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
101 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 ...
29
votes
6answers
3k 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
372 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. ...
17
votes
19answers
3k 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
842 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 ...
40
votes
5answers
4k 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 ...
13
votes
2answers
480 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
151 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 ...
1
vote
1answer
281 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
289 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
462 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
391 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
437 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
95 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
122 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
252 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
90 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 ...
11
votes
4answers
800 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
187 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 ...
7
votes
0answers
234 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
49 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
277 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. ...
19
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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
301 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 ...