**1**

vote

**1**answer

93 views

### Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign:
$vv=Q(v)$ (see, for instance, Wikipedia)
$vv=-Q(v)$ (see, for instance, MathWorld ...

**65**

votes

**10**answers

7k views

### Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending ...

**104**

votes

**18**answers

10k views

### How does one justify funding for mathematics research?

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be ...

**1**

vote

**0**answers

168 views

### In what language do you think when doing mathemematics? [on hold]

Today, nearly all important papers are written in English. People whose mother tongue is not English nevertheless have to learn English if they want to be a mathematician. My question concerns these ...

**4**

votes

**0**answers

73 views

### Is there a name for groups of the form $Sp(1)^n$?

A (compact) torus is a Lie group isomorphic to the product of finitely many circles: $T^n = S^1 \times \cdots \times S^1$. Such groups are extremely important in Lie theory, Differential Geometry, ...

**11**

votes

**1**answer

230 views

### Multiplicative infinitesimals in q-analogs?

Risking to be downvoted, here is a very lightweight question.
In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. ...

**5**

votes

**1**answer

233 views

### Intuitive descriptions of some large cardinals

I was trying to formulate intuitive descriptions of some large cardinals.
Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly ...

**37**

votes

**8**answers

5k views

### Getting nervous refereeing a paper

I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read Refereeing a Paper, but ...

**19**

votes

**18**answers

8k views

### What are some applications of other fields to mathematics?

It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some ...

**121**

votes

**81**answers

96k views

### Do good math jokes exist? [closed]

Have a good joke? Share.
I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)

**1**

vote

**1**answer

345 views

### Doing graph theory after a thesis in pure mathematics [closed]

I've just went through the 1st year of my PhD in France, it is related to Floer Homology. I didn't know what it was really about at that time, I chosed this subject because I thought it would combine ...

**67**

votes

**8**answers

11k views

### How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...

**3**

votes

**0**answers

143 views

### Topology on $\mathcal{C}(X,Y)$ to work with homotopy

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...

**18**

votes

**2**answers

1k views

### Felix Klein on mean value theorem and infinitesimals

This is a reference request prompted by some intriguing comments made by Felix Klein.
In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...

**0**

votes

**0**answers

22 views

### Publishing in math journals without having an adviser [migrated]

Is it possible publishing original work in math research without having an adviser?
I had some thought of matrix multiplication which I posted in this forum which wasn't welcomed that well, I plan to ...

**4**

votes

**2**answers

423 views

### How to find volunteer reviewers?

I am currently in a little dilemma about publishing a result related to general matching.
The dilemma is, that I am not associated to any research institute and thus do not have contact to ...

**5**

votes

**5**answers

553 views

### Important results with one or more than one proof [closed]

Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...

**1**

vote

**1**answer

58 views

### best known bounds for spectral radius [closed]

There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the ...

**132**

votes

**44**answers

77k views

### Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...

**141**

votes

**29**answers

33k 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 ...

**144**

votes

**136**answers

31k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**51**

votes

**47**answers

18k views

### An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...

**106**

votes

**72**answers

16k views

### Most helpful math resources on the web

What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let ...

**82**

votes

**8**answers

9k views

### Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...

**134**

votes

**28**answers

57k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**8**

votes

**2**answers

506 views

### Are there any organized websites for seminar/conference videos?

These days, there are many conference centers and universities recording seminars and conference talks and make them available on the web. Some examples:
http://www.fields.utoronto.ca/video-archive
...

**3**

votes

**2**answers

464 views

### Math and social commitment [closed]

I am a master's student and am looking for ways that link a certain social commitment with serious math. Since I have not found such an overview yet and in order to raise public awareness of such ...

**0**

votes

**0**answers

93 views

### Does the method below provide any advantages?

I completed course intro to numerical method and lately interested and trying to find special way to solve this ODE.I have try and ask in math exchange stack but do not get any respond so decided to ...

**113**

votes

**53**answers

58k views

### Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...

**10**

votes

**1**answer

377 views

### The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...

**64**

votes

**59**answers

15k views

### Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...

**11**

votes

**1**answer

252 views

### Digital physics and “Gandy-like” machines

Various physicists, famously John Wheeler, have asserted that physical information is the central object of study in physics, in the sense that an object or concept is "physically meaningful" if it ...

**7**

votes

**1**answer

768 views

### Should I quit the PhD? [closed]

I am not sure whether this is the right place to post this question.
I am at the end of my seventh year. I won't have funding neither from my department nor from my advisor next year and I do not ...

**26**

votes

**4**answers

2k views

### Is it usual for a referee to heed updated versions on arxiv?

I've put a paper on arxiv one year ago and I've submitted the version 6 to a journal seven months ago.
During these last seven months, I've given several talks about this work, which led me to ...

**7**

votes

**0**answers

241 views

### Partial differential equations outside of academia [closed]

I've seen a number of career/jobs questions on mathoverflow before, so I thought I would ask. Please excuse me if this isn't the best place for this specific question. Lately I've been really ...

**13**

votes

**2**answers

2k views

### What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...

**154**

votes

**67**answers

53k views

### Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...

**1**

vote

**0**answers

55 views

### Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian ...

**10**

votes

**1**answer

816 views

### What is a field [Körper] really?

The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements:
The ...

**20**

votes

**2**answers

2k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**52**

votes

**4**answers

5k views

### Is it possible to have a research career while checking the proof of every theorem that you cite?

A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless they had checked the proof of every theorem ...

**4**

votes

**1**answer

79 views

### Terminology: jointly completely bounded?

This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those ...

**8**

votes

**6**answers

24k views

### What is a satisfactory way to format definitions in Latex?

There are several ways one may format a definition in latex, but each has their problems.
Use the amsthm package, and the usual style for theorems. This will result in everything italicized. It is ...

**-4**

votes

**1**answer

284 views

### Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...

**19**

votes

**3**answers

2k views

### Why are they called Spherical Varieties?

My understanding is if you have a homogeneous space $X = G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then you call $X$ spherical.
Someone ...

**3**

votes

**4**answers

676 views

### Softwares for drawing hyperbolic surfaces , closed, with boundaries or with punctures ?

In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic ...

**62**

votes

**25**answers

5k views

### What could be some potentially useful mathematical databases?

This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a ...

**154**

votes

**11**answers

46k views

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

**-4**

votes

**2**answers

189 views

### If mathematics is logic and intuition, then [closed]

I am just wondering why Mathematics is often defined as The study of Structures, Logic and Numbers which I can concur with but still retain various questions in mind.
I am a postgraduate student of ...

**30**

votes

**8**answers

4k views

### Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following:
1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}...