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I want to mention Selberg's integral, an $n$-dimensional generalization of Euler's beta integral. Selberg published it 1944 in Norwegian in the journal Norsk Matematisk Tidsskrift. Not surprisingly, i …

Existence as a property: You want to find an object that solves a given equation or a given problem. Generalize what you mean by object so that existence becomes easy or at least tractable. Being an o …

The Banach-Ruciewicz problem: Is the Lebesgue measure the only finitely additive measure on the Lebesgue sets in $S^n$ that is invariant under the rotation action by $O(n+1)$ and has total measure $1$ …

Boto von Querenburg wrote a book on general topology, which is one of the standard source in German. According to Wikipedia the name actually stands for the authors Gunter Bengel, Hans-Dieter Coldewey …

The history of good models for spectra might be an example of missed discoveries. To briefly sketch this history: Spectra (in the sense of topology) were introced by Lima in a dissertation under the d …

For the sake of the readers who are not fluent in German, I provide a translation of the German Wikipedia page (link to the revision at the time of posting this answer): Hermann Künneth (1892-1975) w …

Hironaka's proof of resolution of singularities in characteristic 0 might count. His original proof was very complicated (over 200 pages), but today there exist proofs that are reasonably short and ea …

I have to agree with Scott's comment: Every development has its roots. The following three examples are thus only approximations. The first is Riemann's work on the "On the Hypotheses which lie at t …

A few examples: Regarding Grothendieck's theory of schemes: These were called preschemes at first, while the word 'scheme' was reserved for separated (pre)schemes. ( Preschemes and schemes). An exa …

As already remarked by others: If one tries a narrow interpretation of your question, you are asking a lot. You want someone whose specialty is not mathematics to elucidate a mathematical argument in …

The number theory work of Fermat might be an example. He was rather secretive about his methods and much has to be rediscovered later by Euler. This includes Fermat's two-square theorem: It was first …

I want to mention a few examples from homotopy theory: 1) Bott periodicity (1957): This states (in a form) that the homotopy groups of the infinite unitary group $U = colim \, U(n)$ are $2$-periodic …

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even m …

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like t …

A very nice example in my eyes is Serre's proof of Riemann-Roch: Sometimes, you are just not satisfied with existing proofs, and you look for better ones, which can be applied in different situati …