**0**

votes

**0**answers

105 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

106 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 ...

**31**

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

395 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 ...

**13**

votes

**1**answer

969 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 ...

**41**

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 ...

**14**

votes

**2**answers

498 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

156 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

295 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

296 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

476 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

400 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

452 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

101 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

125 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

264 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

92 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

865 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

197 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

**0**answers

268 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

**0**answers

51 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

**1**answer

306 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

**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. ...

**19**

votes

**8**answers

7k 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

**3**answers

316 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

881 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

188 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

408 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

349 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

134 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

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 ...

**17**

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

440 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 ...

**4**

votes

**1**answer

407 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 ...

**4**

votes

**2**answers

546 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 ...

**8**

votes

**1**answer

269 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

770 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 ...

**111**

votes

**23**answers

25k 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 ...

**32**

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 ...

**14**

votes

**5**answers

936 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

120 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 ...

**26**

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 ...

**2**

votes

**4**answers

459 views

### Should all equations which appear in a thesis be numbered?

I was just wondering if there is any sort of consensus on the topic of when to number math expressions.
For example different lines in a proof, these should be tagged or not tagged?

**31**

votes

**6**answers

2k views

### Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...

**12**

votes

**3**answers

1k views

### Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006.
In his talk the first slide he shows has the following written on it:
...

**5**

votes

**0**answers

442 views

### Refereeing a paper containing strong statements about other papers

The title says it all. Suppose you are refereeing a paper where the author A makes strong statements about other papers by a different author B, like: the proof of Theorem 1 in paper [B] is wrong and ...