Penrose’s singularity theorem Roger Penrose won today the Nobel Prize in Physics for the singularity theorem, which at first glance seems to be a result in pure mathematics.
Questions about the theorem:

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*What kind of mathematical technology was used to prove this?

*What ideas did it require that were new at the time?

*Has its interest since then been mainly in physics or has it also led to new mathematics?

There is a related question on Penrose's broader contributions.
 A: Wille's answer is technically true, but he doesn't talk about the historical context of the result. I think that is important for understanding why such a "simple" result is deserving of a Nobel prize.
Sections below are answers to the numbered questions in your question.

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*The tech here is differential topology as applied to Lorentzian geometry. Penrose wrote a book, "Techniques of differential topology in Relativity". It's an amazingly well written account of the maths needed to prove his original singularity theorem. Good luck getting your hands on a copy. In no way do I endorse searching for the book on libgen.

The actual results of this book are not like your Riemannian differential topology. He wasn't the originator of the basics, but he did put them together in a novel way. There are many many results that use the same kinds of techniques that have amazing consequences (see Krolak's cosmic censorship series for example, or Minguzzi's recent work about evaporating black holes).
What separates this kind of study of relativity from the "more normal approaches" is that it is purely kinematic. There are no dynamics involved. In particular Einstein's field equations arn't used. The only physical assumption is an "Energy Condition" which is expressed as an inequality to do with Ricci curvature. Penrose doesn't do a great job of justifying the inequality on physical grounds. For that I suggest you consult Hawking and Ellis which very clearly demonstrate that the inequality follows from some assumptions about how classical matter behaves. The inequality is used in the Raychaudhuri equation to ensure that solutions diverge in finite affine parameter. The inequality is just there to ensure a technicality is true.


*As mentioned the differential topology techniques had been (or were being) developed by a bunch of people. Geroch, Hawking, Ellis, Penrose and a little later Clarke, Kronheimer (and a few others) are the main names. Penrose's insight was to combine the various existing results in an interesting way.

Fundamentally the singularity theorem works like this: Assume that the manifold is maximally extendend, assume a condition that implies that no conjugate points along geodesics can exist, and assume a condition that ensures that if a curve is complete then it has a pair of conjugate points. The resulting contradiction is used to justify that the manifold has an incomplete geodesic.
Beem, Erhlich and Easley presented a super refined version of this argument in Theorem 12.43 of their book. You should check this theorem out. If you read it I guarantee you will first say "WTF" then "but that follows directly from the assumptions". Penrose has been accused of this BTW. Some people have historically claimed that Penrose assumed what he proved. As a philosophical point this is true of all mathematical knowledge, but some how Penrose's singularity theorem is a little "too on the nose".
So what was the reason for the Nobel prize? Where is his original contribution? It comes from the context of physical research back when Penrose published. I know a paper from Senovilla has already been mentioned, but you should read an earlier and much better paper by him: https://arxiv.org/abs/1801.04912 "Singularity theorems and their consequences".
So... there's these three Russians. In 1963 Lifschitz and Khalatnikov publish a paper, (https://www.tandfonline.com/doi/abs/10.1080/00018736300101283) in which they explicitly state, "An attempt is made to provide an answer to one of the principal questions of modern cosmology: ‘does the general solution of the gravitational equations have a singularity?’ The authors give a negative answer to this question.' (That's a quote from the abstract). In 1965 Penrose publishes his paper which proves that Lifschitz and Khalatnikov's paper is bullshit. In 1970 Belinskii, Lifschitz and Khalatnikov (usually write BKL) publish a paper claiming to have demonstrated that generic solutions of Einstein's field equations do have singularities.
Lifschitz and Khalatikov are / were big names. Their second paper birthed the dynamical systems approach to cosmology and one of the more important conjectures in General Relativity (check out https://en.wikipedia.org/wiki/BKL_singularity). These guys work was important.
Their first paper is super complicated analysis of a certain class of solutions to Einstein's field equations that they claimed were generic.
Penrose's theorem is even more generic. As in so generic it has been claimed that the result is self evident. Yet... big wig in the field didn't "believe" in generic gravitational singularities.
In fact, just as Senovilla states in the paper above, Penrose's contribution was convincing astronomers that they should take black holes seriously. Because in in the 60's they didn't. They believed that black holes were a failure of GR and demonstrates that the Einstein's model for gravity was flawed.
That's why Penrose's Nobel prize is jointly awarded with the people who produced the physical evidence for black holes. That's his contribution. It's not actually about the math, it's about a result that changed the whole direction of an area of research. That's why it's a deserving award.


*Well yeah. Absolutely piles of new research and ides. I did my PhD on this stuff, and that wasn't that long ago. I could chat about this literally forever. The big big big question is: Given only the (super minimal) assumptions of the singularity theorems is it possible to provide bounds on curvature divergence? I.e. do the kinematics of gravity imply behaviour as an observer falls into a singularity or do you need dynamics too. I literally mean this: EVERY THING IN CURRENT RESEARCH IN GENERAL RELATIVITY IS ABOUT SINGULARITIES. Or at least can be traced back to it. Penrose's theorem gave GR a reason for existing beyond bragging rights for solving a difficult set of differential equations. For independent justification of this: https://en.wikipedia.org/wiki/History_of_general_relativity#Golden_age.

So yeah... Beyond a more specific question about research I think yelling is ok right?
Edit: Sorry I left of some commentary about mathematical innovation. All singularities theorem look suspiciously like Riemannian rigidity results. So one mathmatical example of the influence of Penrose's work is "The Lorentzian Splitting Theorem", see Chapter 14 of BEE. More generally Penrose's theorem shows that geodesic completeness is a big issue in Lorentzian manifolds. More so since Hopf-Rinow is false. There's been a lot of work on understanding geodesic completeness as a result. There's also been a lot of work on geometrically inspired compactifications of Lorentzian manifolds (unlike the Riemannian case there is no canonical distance). There is an ongoing conference series on Lorentzian geometry. Here is a link to next years one: http://www.uco.es/gelocor/. You could have a look through the list of speakers and their topics to get more of an idea.
Lorentzian geometry is very difficult (due to the failure of basic results - like Hopf-Rinow) and is rife with unresolved issues of completeness. There is also not a great deal of interest in the area among mathematicians.
A: I skimmed the survey article on this by Senovilla and Garfinkle, as recommended by Wojowu. Based on that, and on my studies of the singularity theorems in graduate school, I would say:

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*The basic technology was some compactness, some differential counterparts to cut points and conjugate points, and the tensor manipulations of which Penrose was a virtuoso.


*Senovilla and Garfinkle identify two new ideas in Penrose’s paper: the definition of a singularity as an incomplete timeline or lightlike geodesic; and the idea of a closed trapped surface.


*Senovilla and Garfinkle say: “the most important legacy of the 1965 singularity theorem is the fundamental notion of closed trapped surface....It is not only very useful in the general analysis of gravitational collapse, in the formation of black holes, and in cosmic censorship, numerical relativity and isoperimetric inequalities, it has also become an object of interest for mathematicians —see for instance the use of trapped surfaces to prove the decay rate of gravitational radiation flux — and it has evolved into a richer fauna of interesting ‘trapped-like’ submanifolds with many geometrical and physical implications.”
Their one citation for interest to mathematicians is the article mentioned in between dashes:

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*Mihalis Dafermos, Igor Rodnianski, A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math. 162 (2005) 381-457, doi:10.1007/s00222-005-0450-3, arXiv:gr-qc/0309115
So the interest of the theorem has
indeed been mainly for physicists.
A: 1
Penrose's singularity theorem is a bit of a misnomer.
Penrose never showed that there is a singularity in the spacetime.
What he proved is that the spacetime cannot be timelike or null geodesically complete. As is now well understood, this does not necessarily mean there there is a singularity (in the sense of a region of extreme curvature).
A much better name for the theorem is incompleteness theorem.
2
Beyond basic differential geometric concepts that are covered in common undergraduate courses, such as the concepts of the cut/conjugate points, the key idea that is used is the Raychaudhuri equation for null geodesics, which is a specific form of the Jacobi equation for Jacobi fields along geodesics, but specialized when we consider a family of null (or in the case of the original Raychaudhuri-Laudau equations, time-like) geodesics.
Those of us familiar with the Jacobi equation understands that it says that the rate of acceleration of the separation of nearby geodesics are governed by a curvature quantity. And here is where the theorem is no longer purely geometric: the curvature quantity involved can be related by Einstein's equation to the space-time matter content, and under "reasonable assumptions" this curvature quantity can be assumed to be signed (or zero).
So this means that the presence of reasonable matter will cause nearby null geodesics to want to focus toward each other, similar to how geodesics tend to want to behave on positively curved Riemannian manifolds. So from here we see that there must be some conjugate or cut points that comes up from this focusing.
In terms of lasting mathematical impact, probably this step is the strongest for the modern mathematical GR community. What Penrose demonstrated is that one can pull out monotonicity properties for the evolution equation in a useful way, even though the equations of motion is manifestly time-symmetric. It cemented the importance of thinking about the Raychaudhuri equations (as well as the geometry of null hypersurfaces), and also lends a sort of different philosophy to what is and isn't doable in mathematical GR (this latter is a bit harder to describe).
3
The other main ingredient is a careful understanding of the causal structure of spacetime. By the arguments in the previous step, Penrose showed that the boundary of a certain space-time set is necessarily compact, due to the presence of cut and conjugate points.
A detailed examination of the causal structure of the spacetime, gives a different characterization of the same boundary. Assuming that the space-time is geodesically complete, one can prove from general principles that the same boundary must be a non-compact set.
The contradiction is what leads to a proof of incompleteness.
For someone trained in classical differential geometry, this last ingredient, the understanding of the causal geometry (which is only present in Lorentzian and not Riemannian geometry), is probably the least familiar.
