New Shape Opens ‘Wormhole’ Between Numbers and Geometry
quantamagazine.org
quantamagazine.org
> Mathematics has received a rare gift, in the form of a mammoth 350-page paper posted in February that will change the way researchers around the world investigate some of the field’s deepest questions.
> The work is a collaboration between Laurent Fargues of the Institute of Mathematics of Jussieu in Paris and Peter Scholze of the University of Bonn.
> It opens a new front in the long-running “Langlands program,” which seeks to link disparate branches of mathematics — like calculus and geometry — to answer some of the most fundamental questions about numbers.
> The Langlands program is a sprawling research vision that begins with a simple concern: finding solutions to polynomial equations like x2 − 2 = 0 and x4 − 10x2 + 22 = 0.
> “The Langlands program is a network of conjectures that touch upon almost every area of pure mathematics,” said Caraiani.
Had he also done the same for display managers, we would be fine.
Mathematicians should have been taught to use Scheme syntax instead.
It's also inspiring the way it's written, as opposed to a more academic style.
For academics: is this tone counter productive? For us lay folk: is this tone helpful?
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.
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.
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.
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...
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.
That is one hell of a life.
I wonder if there’s a generic name for these things. Unifying theories? Category theory? We don’t tend to see serious attempts outside of mathematics and physics, which I believe is a missed opportunity.
So many fields seem to be fairly homomorphic, yet very few people seem to notice. For example, electricity transmission, water distribution, telecommunication, transportation networks, and electronic circuits are pretty much identical yet use completely distinct vocabularies. The same is true for apps and multimedia (movie, tv, book, radio, album, game, newspaper, magazine, phone) which first appear to be discrete concepts, but actually exist on a continuous spectrum.
I believe that the complexity of today’s world is merely magnified by our choice of words. A semantic refactoring might just be the best way to improve humanity’s productivity.
Homotopy type theory is more for about foundations of math stuff, i.e. which axioms you use to prove things. ZFC is by far the most common system to work within, not HTT.
[0] ..despite taking years of study to understand the conjectures
https://courses.lumenlearning.com/wmopen-collegealgebra/chap...
Also, where do you see that quote in the article? The only part about circles I saw was the description of the tangent bundle of a circle. This is exactly the kind of example one might see in a first course on differential topology, so I don’t see the issue.