Évariste Galois

It's something called Galois theory.

And one of the young mathematicians who came up with the arguments that led to explaining this, his name was Evariste Galois.

He died in a duel at the age of 20.

This is his big headline fact.

So it's very much having to do with analysis and analytic things.

And there should be some way to make this correspond to representations of Galois groups.

So that is like algebra and number theory.

And of course, you know, it's not just, oh, there's a way of matching things up, right? Like you might think, oh, there's a bijection, right? Well, that's, who cares about bijections, right? That's pretty, that's easy.

And that sort of is like, you know, that's very much related to this whole Langlands and Wiles proof of Fermat's Law theorem.

I see.

So earlier you said that, you know, Langlands program is broadly the correspondence between, you know, the representations of reductive groups and the Galois groups.

So is there anything that we learned about any of these ingredients using this correspondence that was hard to learn without using this correspondence?

But for degree 5, there's no such formula.

And the reason was found by a French teenager, Averiste Galois, who invented this whole way of thinking about symmetry.

The night before the duel that killed him at 20, he wrote his ideas down in a letter.

And in today's language, the reason is that atom of 60, the general equation of degree 5, carries it inside its symmetries.

You know, French lefties have a sexy image.

In the old days, they'd smoke Galois before smoking was not a thing and, you know, sip black coffee on the left bank and so on.

In fact, the French left is among the most Stalinist, ruthless, intolerant, and vicious political forces in Europe.

You know, they educated all the great totalitarian dictators of the 20th century, Pol Pot and in Cambodia, Ho Chi Minh in Vietnam.

Maybe I was meant to follow Ramanujan.

And so I started to work on a thesis based on his work, and I'm so glad I made that choice because by the end of my PhD, uh, what I worked on was called the Theory of Galois Representations, which was meant to study Ramanujan's backwater mathematics.

But by 1993, the bombshell news in mathematics for the end of the 20th century was a proof of Fermat's Last Theorem, and the proof of Fermat's Last Theorem depend on these Galois representations.

I don't know what it is.

Following Ramanujan every time he appeared has been like the best decision I've ever made in my life, and every one of those could have gone a different way.

Well, and this, this, by the way, is something that has always been-- Like, I came from a math background.

Like, I thought I was going to be a mathematician, and I was very inspired by mathematicians like Galois or Gauss, right? These people who were working on like 100, 400-year time horizon, right? And I think there's something deeply beautiful about mathematics being about sort of understanding the structure of the universe.

And so it's not just about-- And there's always these questions about, is math invented or discovered? And that in my mind, it's something that the, the beauty and the meaning of mathematics, it's not something that is objective in some sense.

It's, it's in the eye of the beholder.

Maybe I was meant to follow Ramanujan.

And so I started to work on a thesis based on his work, and I'm so glad I made that choice because by the end of my PhD, uh, what I worked on was called the Theory of Galois Representations, which was meant to study Ramanujan's backwater mathematics.

But by 1993, the bombshell news in mathematics for the end of the 20th century was a proof of Fermat's Last Theorem, and the proof of Fermat's Last Theorem depend on these Galois representations.

I don't know what it is.

Following Ramanujan every time he appeared has been like the best decision I've ever made in my life, and every one of those could have gone a different way.