**7**

votes

**2**answers

222 views

### Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...

**3**

votes

**3**answers

293 views

### Survey papers of results (relevant to mathematics) produced during research on the relationship between mathematics and music theory

It is well-known that there is a relationship between music and mathematics, and there are many references that explore this topic (for example, Benson's book).
However, I would like to ask if there ...

**1**

vote

**0**answers

98 views

### Why are they called 'pernicious' numbers?

The definition of a pernicious number:
In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.
The meaning of ...

**8**

votes

**1**answer

442 views

### Open access journals in number theory

I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my ...

**3**

votes

**1**answer

407 views

### Do hom-sets really live in the category Set?

This isn't really a research-level question (sorry!), but I asked on
math.se (link), and though the question was upvoted a few times, I didn't
get any answers. So since there may well be more ...

**16**

votes

**8**answers

913 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**0**

votes

**1**answer

247 views

### What would undergraduate research consist of? [closed]

For undergraduate mathematics students looking to go to grad school, what kind of opportunities are open for research? I would assume undergrads would not be under any pressure or obligations to write ...

**2**

votes

**1**answer

159 views

### Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
...

**-3**

votes

**1**answer

231 views

### How many of Ramanujan's discoveries have had a practical application? [closed]

I was reading about the Indian mathematician Srinivasa Ramanujan who, before dying at the age of 32, independently compiled nearly 3900 results (this is from Wikipedia). So based on this he seems to ...

**21**

votes

**1**answer

626 views

### Which properties of a variety are detected by its derived category of coherent sheaves?

Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about ...

**11**

votes

**2**answers

633 views

### What is the longest recorded gap between “proof” of a “theorem” and discovery that the result is false [duplicate]

I hope this question is not a duplicate. I am motivated by wondering when widely accepted results may be considered have a secure place in the mathematical literature. The question is intended to ...

**-5**

votes

**2**answers

333 views

### Are there any with Erdös number 1 on mathoverflow? [closed]

I learned today that one of my professors wrote a paper with Erdös and Sós back in 1985, thus granting him the honor of having his Erdös number equal 1.
I was wondering - did any of the ...

**0**

votes

**1**answer

111 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

**3**answers

648 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.

**8**

votes

**1**answer

721 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 ...

**51**

votes

**5**answers

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

**0**answers

231 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 ...

**13**

votes

**2**answers

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**

votes

**7**answers

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

**5**answers

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

**5**answers

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

**2**answers

408 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

**1**answer

181 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

**1**answer

275 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

**3**answers

931 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

**2**answers

483 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

**1**answer

347 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

**1**answer

239 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

**0**answers

359 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

**1**answer

162 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

**3**answers

2k 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

**7**answers

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

**1**answer

168 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
...

**5**

votes

**0**answers

256 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

**1**answer

752 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

**0**answers

99 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

**0**answers

103 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 ...

**30**

votes

**6**answers

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

**0**answers

382 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

**19**answers

4k 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

**1**answer

899 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

**5**answers

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

**2**answers

490 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

**0**answers

153 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

**1**answer

287 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

**4**answers

292 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

**12**answers

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

**2**answers

465 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

**3**answers

397 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

**0**answers

443 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 ...