It was well written but I wish being a geometry related topic there were some figures to help with the explanation. I’d like to /see/ examples of how this all works to help me understand.

Unfortunately the "geometry" involved is not the kind of geometry that lends itself towards being easily drawn to give understanding.

If you had to draw a picture of it, it would just be a sphere (since it shares many properties with a genus 0 complex curve).

But it's a very different beast than usual geometric shapes. A lot of modern number theory is motivated by an analogy between "p and t", where t is a parameter in a polynomial like t^2 - 2, and where p is a prime number. This space takes the analogy to an extreme where p literally is the parameter for the space[0], like t would be a parameter for the real line.

[0]Not exactly true - true for the "disk" cover of the FF curve.

Do you have a few years to spare? Algebbraic geometry is immensely involved even by grad level standards. You cannot really jump from college under grad math to algebraic geometry in the way you can for other areas. You would need to start by having a good grasp of linear algebra.

Yeah I’m really looking forward to the Standup Math or/and (preferably and) 3blue1brown version.

I would be surprised if this gets a 3blue1brown episode - this branch of mathematics takes a lengthy time to wrap one's head around. 3blue1brown topics tend to show up in undergraduate math, whereas the Langlands program (and the Fargues-Fontaine curve in particular) would really only be studied by a subset of graduate students studying Number Theory.

As a noob I'd say this also falls into that category: [0].

Who knows, if enough people ask about it on his reddit (topic requests) :) : [1]

[0] https://www.youtube.com/watch?v=mH0oCDa74tE

[1] https://www.reddit.com/r/3Blue1Brown/comments/mllj2s/topic_r...

Yes! I just came here to say this too.

For academics: is this tone counter productive? For us lay folk: is this tone helpful?

As more of a layperson, I have mixed feelings. As science journalism goes, it's excellent, but reading this sort of frontier news feels like a waste of time to me now. I only really found out that progress is being made in fields I don't understand. I bothered to check this article out because multiple people posted that it's very good, but the time I spent reading it would've been better put into getting a little bit more real understanding of old related discoveries like the unsolvability of the quintic, which is within reach at my level.

I think that's more about the content than the writing. This is an incredibly esoteric topic which is only really of professional relevance to maybe a couple of hundred people in the world.

Given that, I think the writing did an excellent job of making it sound interesting and almost accessible.

In comparison quintics are a two hundred year old puzzle and not a two year old puzzle, so it's easy to find explanations elsewhere.

Yes. I have a couple of relevant books checked out from the library, actually.

But I wish I hadn't sort of defaulted to reading way too much pop science over the decades. Diminishing returns set in long ago.

As someone who is maybe somewhere in the middle (I have a Masters in Maths, but haven't used it in earnest since graduating ~10 years ago) - I found this spot-on. Engaging enough to pull me in, technical enough to give some idea of what exactly is going on, but not too far in either direction.

Read a similar article from this publication on quantum mechanics, and have a similar background in that area, and I concur completely. Maybe that's just the exact audience this is targeted to but it's by far the best "popular-press" publications on subjects of this level I've read.

Depends on what you mean by counter productive. You wouldn't cite Quanta Magazine if you were writing the next journal article on this topic intended for peer review. Instead, you would go to the original paper "Geometrization of the local Langlands correspondence" and pull that apart to look for ways you can improve upon what has already been done.

But mathematics is a big area (there is a reason why it's a plural: it's better to think of it as more of a family of topics than a single science.) And if you're a mathematician from an unrelated branch of mathematics, an article like this is written in a way where you can understand most of the underlying concepts reasonably intuitively.

There is nothing that can help laypeople if the goal is to understand the math. I mean the official harthstone text is hundreds of pages of hard grad math.this was the 70s .a lot has transpired.

I need more pictures. I'm obvious a layperson.