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I'm not at Jane Street, but at G-Research (also quant finance). Much as a joke goes which I first heard in the Effective Altruism sphere, our quants come from all kinds of backgrounds: their PhDs are in fluid mechanics, statistical physics, pure maths, you name it!

how easy do they switch to Quant finance? I don't know anything at all but it seems like an entire new field to learn.

Yes.

The one small firm that everyone in the industry has heard of is basically all phd guys screwing around with theory and execution and getting filthy rich in the process.

How about design or other backgrounds such as hci?

Yes.

What do the math or CS PhDs do at Jane Street? I don't know much about trading so I am curious.

If you're interested enough, read Gregory Zuckerman - "The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution" (2019)

The first half of the book goes over the academic history of quant trading.