31

It seems (as mentioned by Sam Hopkins above) that the Singularity Theorem is the official reason for the Nobel Award. But that is by no means the only (and perhaps not even the most important) contribution of Sir Roger Penrose to mathematical physics ( not to mention his works as a geometer and his research on tilings, and so many other things). In Physics, ...


22

A very interesting contribution (not directly related to relativity) is joint with Moore on the so-called Moore-Penrose inverse or generalized inverse, which is crucial in inverse problems theory and ill-posed problems.


22

I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete*, but just some stuff off the top of my head.) 1 The "black hole" theorem (incompleteness theorem) is closely related to, yet subtly different from, the Hawking-Penrose Singularity Theorems. The ...


16

In general, I love the Wayback Machine, so when I read your question, I was very in-favor of it. But, we should also think about unintended consequences. If every random mathematical note one uploads to a personal webpage (including notes for students) gets archived somewhere, then that might have a chilling effect on peoples' willingness to upload such ...


14

I would say Penrose is a mathematical physicist and I don't think he can be considered (at least not primarily) to be a pure mathematician. For example, his argument for the Penrose inequality is a plausible but non-rigorous physical argument. The main contribution of Penrose and Hawking and the one cited was that they showed (roughly speaking) that if one ...


13

In his book An Introduction to Combinatorial Analysis, Riordan observed that the number of ways to choose $k$ objects from $n$ objects, allowing repetition and disregarding order, can be written $(-1)^k{-n\choose k}$, while ${n\choose k}$ is the number of ways without repetition. This was the first inkling of the vast subject of combinatorial reciprocity. ...


9

In my previous answer, I focused on unintended consequences of creating a Wayback Machine for math. But I'm worried that answer is preventing us from answering the actual question. Would it be feasible to set up something like the Wayback Machine but specifically targeted at mathematics, so that we could ensure higher quality preservation than the actual ...


7

Ramsey Theory has to be mentioned in this context I think. This is a somewhat obscure but interesting branch of combinatorics that is named after the mathematician/philosopher Frank Ramsey who proved its first result through Ramsey's theorem. Interestingly, Ramsey only proved this theorem in passing as a minor lemma. He was actually trying to prove a ...


7

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, it did not get a lot of publicity there. Later it was key to results in random matrix theory and other areas. There is an excellent article by Forrester and ...


5

Neural networks are a great example right now in machine learning. They were around for decades before the computing power to actually train them properly became available.


5

The following quote is from Joseph Dauben (in his article Georg Cantor: The Personal Matrix of His Mathematics, Isis, Vol. 69, No. 4, Dec., 1978, pp. 534-550). When he suddenly suffered his first breakdown in May 1884, Cantor had just returned from an apparently successful, quite enjoyable trip to Paris. He had met a number of French mathematicians, ...


5

The answer to the question can be found in Section I (especially p.11) of this source (it is a newly typeset version of Joseph E. Quinsey's 1980 Oxford doctoral thesis Some Problems in Logic: APPLICATIONS OF KRIPKEā€™S NOTION OF FULFILMENT).


4

Richard Stanely's 1973 paper "Linear homogeneous Diophantine equations and magic labelings of graphs" was the first time commutative algebra was used to study convex polytopes. But the paper is not really about polytopes per se. Rather, its main focus is on resolving the Anand-Dumir-Gupta conjecture about "magic squares," specifically, ...


3

If memory serves, in James Gleick's book Chaos, he describes the origins of this field as attempts to find numerical bugs and rounding errors in PDE solvers -- before it was realized that something far more profound was happening.


3

I think the implicit function theorem fits very well. The idea of solving an implicit equation is simple, an for examples like the circle one might call it a small idea. However, the implicit function theorem is still very useful and can be applied in various situation, for example to prove existence in complicated situations.


2

As the others point out, there is no knowing about the earliest this was written down. For example, the notion of regularity of a ternary form is due to Dickson, but universality is an easier concept and could have gone without any name for quite some time. Worth pointing out that all universal ternaries can be described. Three out of four types are in ...


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