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Re:Imperial mathematician scoops $3M Breakthrough Prize by melling on Friday September 11, 2020 @08:05AM (#24441761)
here’s his paper that won him the prize. It’s long at 180 pages.

https://arxiv.org/abs/1303.5113

Wish there was something between a Quanta article and the paper that described his thinking.

Re:Imperial mathematician scoops $3M Breakthrough Prize by kkylin on Friday September 11, 2020 @12:54PM (#24444695)
A quick search shows this:

https://arxiv.org/abs/1401.3014

which still requires a substantial mathematical background, but seems rather less technical (I've not read it). There's also a textbook on rough paths by Friz and Hairer with an introduction to regularity structures; if you're at an academic institution with access to SpringerLink, you may be able to get it easily.

Finally, Martin gives lots of talks, and he's an exceptionally clear speaker. I wouldn't be surprised if there's something on YouTube that's reasonably accessible to someone with a solid math / physics background.