Here is a story I heard many years ago, and have no confirmation of: Apparently, there was Asst Professor X at a provincial department Y, and he was up for tenure. Professor X's advisor was a famous Japanese mathematician Z at an Ivy League school. Naturally, he was asked for a letter, which he ...

I am not sure where or when this happened, but I still think there may be some truth to the story. Once someone from the engineering (or physics?) department of some university came to see Joseph Bernstein and asked if he knew a formula for a conformal mapping of the interior of a regular $n$-go...

I have heard (from two sources) that at the University of Chicago a senior faculty member was temporarily banned from teaching undergraduate courses. The reason is that during a first semester undergraduate linear algebra course he did everything over the Quaternions. This one isn't so much acad...

I heard this in Oxford in 1970. I can't believe it: A PhD student decides to see what happens if he assumes the inverse of the triangle inequality. He finds he can prove that there are various interesting consequences - for instance, certain sets of points must be collinear. He eventually writ...

Here's my list of false beliefs ;-): If $U$ is a subspace of a Banach space $V$, then $U$ is a direct summand of $V$. If $M/L, L/K$ are normal field extensions, then the same is true for $M/K$. Submodules/groups/algebras of finitely generated modules/groups/algebras are finitely generated. The ...

Weierstrass' function is an example of a function that is continuous, but nowhere differentiable, and can be visualized as being "infinitely wrinkled". I'm having trouble, however, imagining how the integral of such a function would appear. All the techniques that I know of for approximating func...

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded and continuous almost everywhere, which the Dirichlet function satisfies. On the other hand, th...

Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by $$f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \\ 0 & x = 0 \\ \\ \frac{1}{q_x} & x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \ q_x \in \mathbb{N}, \ \text{gcd}(|p_x|, q_x) = 1 \end{cases}$$ I...

I am trying to solve this exercise from Royden's 3rd edition. The question is as follows: Let $f$ be a real-valued function defined for all real numbers. Show that the set of points at which $f$ is continuous is a $G_{\delta}$. Let $$A_n = \{y : \text{there is a }~\delta_y \gt 0 : |f(s)-f(...

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while they are learning mathematics, but quickly abandon when their mistake is pointed out -- and also in ...

It's a long list of wonderful answers already, but I can't resist... Question: Is it possible to find six points on a square lattice that form the vertices of a regular hexagon? Proof without words: Hint: A square lattice is invariant under rotation by π/2 around any lattice point. Use redu...

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list in my teaching, to motivate general education undergraduates, and early year majors, suggesting t...