I received an email today about the award of the [2020 Nobel Prize in Physics](https://www.aip.org/science-news/nobel2020) to **Roger Penrose**, **Reinhard Genzel** and **Andrea Ghez**. [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose) receives one-half of the prize "for the discovery that black hole formation is a robust prediction of the general theory of relativity." Genzel and Ghez share one-half "for the discovery of a supermassive compact object at the centre of our galaxy". [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose) is an English mathematical physicist who has made contributions to the mathematical physics of general relativity and cosmology. I have checked some of his works which relate to mathematics, and I have found the paper


* M. Ko, E. T. Newman, R. Penrose, *The Kähler structure of asymptotic twistor space*, Journal of Mathematical Physics **18** (1977) 58–64, doi:[10.1063/1.523151](https://doi.org/10.1063/1.523151),


which seems to indicate Penrose has widely contributed generally to the mathematics of general relativity like tensors and manifolds. Now my question here is:



> 
> **Question**
> What are contributions of Sir Roger Penrose, the winner of the 2020 Nobel prize in physics, to the mathematics of general relativity, like tensors and manifolds?
> 
> 
> 


We may motivate this question by adding a nice question which is pointed out in the comment by Alexandre Eremenko below where he asks: *Is Sir Roger Penrose the first true mathematician to receive a Nobel prize in physics?* If the answer is yes, then Sir Roger Penrose would say to us "before being a physicist you should be a mathematician". On the other hand, in my opinion the first mathematician to be awarded several physics prizes is the American mathematical and theoretical physicist [Edward Witten](https://en.wikipedia.org/wiki/Edward_Witten). This seems to meet Sir Roger Penrose in his research such as cosmology and research in modern physics (Einstein general relativity).


**Related question**: [Penrose’s singularity theorem](https://mathoverflow.net/q/373423/51189)



What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?
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, his grand idea is **Twistor Theory**, an ongoing project which is still far from completion, but that has been incorporated in other areas (see for instance [here](https://royalsocietypublishing.org/doi/10.1098/rspa.2017.0530) for its connection to Strings Theory, and also there is another connection with the Bohm-Hiley approach using Clifford Algebras, see [here](http://www.bbk.ac.uk/tpru/BasilHiley/PS16Twistors.pdf) ).


But his influence goes even beyond that: Penrose invented **Spin Networks** in the late sixties as a way to discretize space-time. The core idea was subsequently incorporated in the grand rival of String Theory, **Loop Quantum Gravity**. As far as I know, all approaches to a background independent Quantum Theory of gravity use spin networks, one way or the other.


Moral: Congratulations Sir Roger !


**ADDENDUM** @TimotyChow mentioned that my answer does not address the ask of the OP, namely Penrose's contribution to General Relativity. I have mentioned two big ideas of Penrose, namely Spin Networks and Twistor Theory. The first one is, as far as I know, not directly related to standard relativity, rather to "building" a discrete space-time. It is not entirely unrelated, though, because the core idea is **that space-time, the main actor of GR, is an emergent phenomenon**. The ultimate goal of spin networks and also of all theories which capitalize on them is to generate a description of the universe which accommodates Quantum Mechanics and at the same time enable the **recovery of GR as a limit process**.


As for the second theory, Twistors, I am obviously not the right person to speak about them, as they are a quite involved matter, with many ramifications, from multi dimensional complex manifold to sheaf cohomology theory, and a lot more.


But, for this post, I can say this: the core idea is almost childish, and yet absolutely deep. Here it is: Penrose, thinking about Einstein's universe, realized that **light lines are fundamentals, not space-time points**. Think for simplicity of the **projective space**: you reverse the order. Rather than lines being made of points, it is points which are the focal intersection of light rays. The set of light rays , endowed with a suitable topology, make up **twistor space** (it is a complex manifold of even dimension).


Now, according to Penrose, relativity should be done inside Twistor Space, and the normal space-time can be recovered from it using the "points trick" and the Penrose mapping which transforms twistor coordinates into the lorentzian ones. What is more is that twistor space provide some degree of freedom for QM as well. How? well, think of a set of tilting light rays. Rather than a well defined space-time point you will get a "fuzzy point". But here I stop.


