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For concreteness, consider the $abc$-inequality with $\epsilon=1$, i.e. that there is some integer $K$ such that for all coprime integers $a,b,c$ with $a+b=c$, one has $$abc\leq K \prod_{p|abc} p^6.$$ …

The error in the paper is in the proof of Theorem 7.2. The proof of Theorem 7.2 is immediately suspicious because of how vague it is in places and because of how lofty the expository text before and a …

Note that $a_k$ is always a multiple of $k$ (in particular, this structure disrupts the random model proposed in comments). We can exploit this structure to simplify the recurrence and clarify the dy …

(This is a variant of Cartier's argument mentioned by Dan Ramras.) Let $X$ be a finite set of size at least $2$. Let $E$ be the set of edges of the complete graph on $X$. The set $D$ of ways of dire …

To answer the question in the title: "No." And I would imagine that Ukrainian mathematicians would boycott any ICM held in Russia, in these times. So the question of whether Russia would honor their p …

I think that a large part of the difficulty we have in understanding why this result is considered important is that it is psychologically difficult to put oneself into the shoes of mathematicians of …

Over decades, and across multiple research fields, I've noticed a way to predict I'm on track to make progress. I discover something interesting, only to learn it is already known. As a student, this …

Some mathematicians seem to agree with you, and strive only to state and prove the most general versions of their theorems. I've had co-authors express that view. And I've sometimes had referee report …

I apologize for writing a lengthy answer, but I get the feeling the discussions about foundations for formalized mathematics are often hindered by lack of information. I have used proof assistants for …

In a letter to Frobenius, Dedekind made the following curious observation: if we see the multiplication table of a finite group $G$ as a matrix (considering each element of the group as an abstract va …

How about a lecture on proof assistants/formal proofs? Most mathematicians are still skeptical of the value of proof assistants, and it's certainly true that proof assistants are still very difficult …

There is a beautiful article (a write-up of a talk, actually), by E.M. Purcell, Life at low Reynolds number, that explains how bacteria swim. Low Reynolds number is the technical way to phrase the sta …

$\DeclareMathOperator\cr{cr}\DeclareMathOperator\pcr{pcr}$For the pair crossing number $\pcr(G)$, the short answer is yes the crossing lemma holds for drawings on the sphere, but it is not known wheth …

Mathematical notation in a given mathematical field $X$ is basically a correspondence $$ \mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$ between mathem …

There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet …