Sizable article - write down a few highlights about the Langlands program.
> 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.
That’s “x² − 2 = 0” and “x⁴ − 10x² + 22”. We should all take more care to transcribe exponents correctly when copying between media. (Actually, we should go back in time and prevent anyone from inventing superscripts and subscripts entirely, but that is a longer–term goal.)
This has been bothering me since I basically started using computers: why are default inputs for computers so bad for writing math? Computers are basically bathed in the field of mathematics and yet writing math with a computer is quite an unpleasant experience. Why?
Too many years of inadequate display technologies (text only) plus some powerful notations which could describe math to computers in other ways (using functions like sqrt() and pow()). Also mathematicians like to invent typographically hard to use symbols, which would need to be widely implemented first before they can publish their research, so programmers only make specialised packages for mathematicians instead of system-wide support.
The notation is far too complex and ambiguous, with too much reuse of notation to mean different things in different subfields. For example, superscript numbers are used for exponents, and subscripts are for indices, right? Well except for certain cases where vectors are indexed by superscripted numbers instead.
Mathematicians should have been taught to use Scheme syntax instead.
I enjoyed the way this article was written - it sounds like the synopsis of a new TV show on Showtime or Netflix or something (of course based in a stylized post-war Europe or similar with muted colors).
It's also inspiring the way it's written, as opposed to a more academic style.
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.
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 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.
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.
It looks similar to Homotopy Type Theory, which also attempts to connect seemingly unrelated fields of mathematics into one.
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.
The Langlands program is not similar to homotopy type theory. It is 'unifying' only in the sense that it makes striking, deep, yet very concrete[0] connections between 2 seemingly disparate areas of math, and because of this the tools needed to make progress in it end up touching all areas of mathematics.
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
I'd love to know how these mathematicians build intuition about multi-dimensional objects. These theoretical mathematicians blow my mind every time I try and read something about them.
Sigh...here we go again. "algebraic geometry is like plotting the equation of a circle" this description and definition is as helpful and accurate as saying a circle is approximated by a square. Pop math writers need to stop likening something as complicated as algebraic geometry to high school algebra.
What makes you say that? The circle is one of the few familiar objects from grade school geometry which is also an algebraic variety.
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.
> 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.