If one tried to pick a single most talented and influential figure of the past 100 years in each of the fields of pure mathematics and of theoretical physics, I’d argue that you should pick Alexander Grothendieck in pure math and Edward Witten in theoretical physics.Several comments give some good reasons for disagreeing with this assessment.
Grothendieck is almost completely unknown outside Mathematics, as his work was in the abstract foundations of algebraic geometry. As for Witten:
And he [Ed Witten] rarely came to our floor, fourth floor, but here he was, coming and knocking at my door, and then saying, “Have you heard about the revolution?”…Witten convinced everyone of these string "revolutions". This one was a minor technical result in 1984. There is still no known relation to the physical world.
I said, “What revolution?” He said, “The SO(32) revolution.”
This "revolution" terminology stems from philosoher T. Kuhn, who based it on a study of the "Copernican revolution", where the Earth does revolutions about the Sun. He said that Copernican theory was not measurably better than Ptolemaic (Earth-centered), but was great anyway because it became accepted.
The lesson here is that if you call something a revolution and persuade your colleagues, you can be a great genius without showing any measurable advantages.
Woit credits Witten largely because he was-influential in conning everyone into studying string theory, a big dead end. Dirac, Feynman, Weinberg, and all the other theoretical physicists just advanced the state of the art, and did what others might have done later. Maybe no one would have bothered with string theory, if it were not for a few leaders like Witten.
The problem with theoretical physics IS pure mathematics.
The universe isn't pure Math observationally, isn't made of math structurally or logically, and isn't informed by the ideological conceit of many mathematicians that depends upon Math existing unto itself apart from physical reality in a platonic alternative realm overlay of some kind that mathematicians mystically can tap into like a collective consciousness...if they do enough drugs and smoke lots of pot. How convenient for them.
A paradigm shift is just a fancy way of saying a whole bunch of toady academic peers changed their group mind en mass when their favorite pet theories fell out of favor, usually because someone died or someone more charismatic came along. If it wasn't for intellectual groupthink and lots of other people's money, there wouldn't even be an academia to lob petty door prizes at.
Academic life revolves around peer hierarchy politics, not functional reality.
“Science advances one funeral at a time.”
― Max Planck