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This is a request for help (with examples, as described below) with a talk I giving to graduate students regarding the dynamics of mathematical research among mathematicians and the development of key ideas towards the solution of open problems.

So, during the talk, I am intending to supply several examples/instances where mathematicians almost simultaneously arrived at the same/similar/different solution, idea, or proof strategy to a particular problem (probably well known or difficult problem). The context of “almost simultaneous” is not with respect to the publication dates necessarily but, give and take, within a couple of years between such publications; more relevant is that the key idea generation/discovery would have been just about the same time.

Naturally, the development of calculus (Newton and Leibniz) is in such a spirit. So also is the proof of the Prime Number Theorem, both the non-elementary (de la Vallée Poussin and Hadamard) and the elementary (Selberg and Erdos). A recent one, I want to believe, is the bounded prime gaps (Zhang and Maynard). Especially, it would be nice if you could also provide a (his)story/link/reference of the account. I am not actually requesting for controversies or issues surrounding the solutions....but the examples of such problems and their solutions.

It is also welcome cases, where such examples may be obscure, or cases where the authors perhaps collaborated (such as, I believe, Gowers and Maurey on the Unconditional Basic Sequence Problem). Little-known or Word-of-mouth examples are also welcome, as long as it is not a private matter (By this, I have in mind, for instance, that I once heard/read somewhere that a quite prominent mathematician had also been developing a proof to FLT along the same/similar lines as Wiles but (perhaps) did not pursue further after Wiles’ announcement and eventual publication. Such instances may be private rather than little-known as the said “prominent” mathematician neither published or intended for their admission to be made public).

PS: This is my first time asking a question here. I do not know if such a question has been asked here before. Will welcome a link to it if so. Also, I don’t know if this is tagged differently or anything of that sort, so will be grateful if it can be made as such if necessary.

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    $\begingroup$ This might be too narrow of a topic, and too spread out temporally to be what you want, but for several independent discoveries of a certain Baire category proof that most $C^{\infty}$ functions are nowhere analytic, see IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC in this 9 May 2002 sci.math post. $\endgroup$ May 24, 2022 at 10:44
  • $\begingroup$ Many thanks @DaveLRenfro. I’d take a look and glean from it a useful example to add to what I have. $\endgroup$ May 24, 2022 at 11:00

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An example of a near-simultaneous discovery which I came across recently involves Liouville's approximation theorem - that for an irrational, algebraic real number $\alpha$ of degree $n$, there are only finitely many rational numbers $p/q$ such that $| \alpha - p/q | \lt 1/q^{n+\epsilon}$ - used by Liouville to generate transcendental numbers. Liouville suspected that the result was not optimal and a number of improvements followed (Thue, in 1909, reduced $n$ to $\frac{1}{2}n+1$ and Siegel, in 1921 reduced $\frac{1}{2}n+1$ to $2\sqrt{n}$). Then, in 1947, Freeman Dyson and Aleksandr Gelfond independently reduced $2\sqrt{n}$ to $\sqrt{2n}$. (It was only in 1955 that Roth published the optimal result by reducing $\sqrt{2n}$ to $2$.)

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A recent example is Conway's Angel problem that was first published in 1982, and four(!) independent solutions was found in 2006 almost simultaneously.

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