**4**

votes

**1**answer

183 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 ...

**25**

votes

**7**answers

3k 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

181 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
...

**6**

votes

**0**answers

537 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

797 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

116 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 ...

**36**

votes

**6**answers

4k 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 ...

**9**

votes

**0**answers

472 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. ...

**18**

votes

**19**answers

7k 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 ...

**15**

votes

**1**answer

1k 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 ...

**51**

votes

**5**answers

6k 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 ...

**14**

votes

**2**answers

533 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

159 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

329 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

312 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 ...

**4**

votes

**2**answers

507 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

422 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

471 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

**1**answer

131 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

**0**answers

139 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

**1**answer

348 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

**1**answer

102 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

**4**answers

1k 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

**0**answers

212 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 ...

**8**

votes

**0**answers

355 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 ...

**4**

votes

**1**answer

370 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 ...

**13**

votes

**3**answers

2k 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. ...

**22**

votes

**8**answers

8k 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 ...

**6**

votes

**3**answers

369 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 ...

**70**

votes

**25**answers

10k 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 ...

**13**

votes

**2**answers

948 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

**1**answer

194 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

**0**answers

438 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

**0**answers

446 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 ...

**2**

votes

**2**answers

142 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, ...

**25**

votes

**9**answers

2k 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 ...

**19**

votes

**7**answers

1k 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 ...

**-4**

votes

**1**answer

455 views

### Publishing problem [closed]

First, I want appreciate your work on this platform, as I have been getting very helpful advice even though I am not a member. My problem is that I have been working on-off on a famous math problem ...

**5**

votes

**1**answer

608 views

### Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...

**5**

votes

**2**answers

669 views

### Why considering schemes over discrete valuation rings?

For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and ...

**9**

votes

**1**answer

465 views

### Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...

**11**

votes

**1**answer

2k views

### ICM 2014 streaming video

Is there a possibility to watch ICM 2014 opening ceremony and the big talks online?
I hope there is since it was possible for the previous meeting.

**5**

votes

**2**answers

861 views

### Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...

**139**

votes

**27**answers

32k views

### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

**34**

votes

**1**answer

3k views

### A topologist is not a mathematician - a small question

Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the ...

**15**

votes

**5**answers

1k views

### Examples of research on how people perceive mathematical objects

What examples are there on research related to human perception and mathematical objects?
For example, the shape of a beer glass influences drinking habits,
since people are bad at integrating.
...

**0**

votes

**0**answers

129 views

### Looking for rapidly converging series for the reciprocal gamma and/or gamma function

There are rapidly converging infinite series for Pi and the such but it is difficult to locate those for either the gamma or reciprocal gamma function. I am searching for rapidly converging series ...

**28**

votes

**15**answers

5k views

### Examples of famous 'workhorse' theorems

I use the term 'workhorse' to describe a theorem which is technically challenging to prove, perhaps very deep, but the statement is either uninteresting at first glance or too imposing to be ...