12 ms·

New Shape Opens ‘Wormhole’ Between Numbers and Geometry

193 points by theafh 5y ago

jesuslop 5y ago
It's about Scholze work on perfectoid spaces. A nice intro talk in Spanish is at https://cienciastv.org.mx/2020/08/el-mundo-perfectoide-adrian-zenteno/ https://cienciastv.org.mx/2020/08/el-mundo-perfectoide-adria...
- infogulch 5y ago
  Didn't Kevin Buzzard work on a proof related to perfectoid spaces using Lean theorem prover that featured here a few months ago?
  - jesuslop 5y ago
    Yep, you are right. Kevin Buzzard, Johan Commelin, and Patrick Massot, slide 31 at https://robertylewis.com/files/lean_intro_slides.pdf https://robertylewis.com/files/lean_intro_slides.pdf
rohanphadte 5y ago
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.
- db48x 5y ago
  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.)
  - Aerroon 5y ago
    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?
    - darig 5y ago
      Because you didn't manufacture the keyboard you think you want? Enjoy your 3 customers. The rest of us aren't bother to learn the alt codes.
    - yetihehe 5y ago
      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.
    - nabla9 5y ago
      Because Don Knuth only invented digital typesetting he had time to only write TeX and Metafont. Had he also done the same for display managers, we would be fine.
    - db48x 5y ago
      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.
hammock 5y ago
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.
- austinheap 5y ago
  Yes! I just came here to say this too. For academics: is this tone counter productive? For us lay folk: is this tone helpful?
  - scubbo 5y 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.
    - gipp 5y ago
      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.
  - abecedarius 5y ago
    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.
8bitsrule 5y ago
This page shows how graphing polynomials can lead to factoring solutions. https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/graphs-of-polynomial-functions/ https://courses.lumenlearning.com/wmopen-collegealgebra/chap...
paulpauper 5y ago
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.
- kevinventullo 5y ago
  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.
- langlands 5y ago
  This is actually the first pop math article I've read in a while that didn't make me roll my eyes.
miguelrochefort 5y ago
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.
- mycall 5y ago
  Graph theories are used for all of your cited examples to explain how they work. They are well studied group homomorphic feature kernels in algebra.
- meowkit 5y ago
  I believe these are referred to as formal theories and are sort of the root connections between philosophy/epistemology and maths.
- langlands 5y ago
  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
AndrewKemendo 5y ago
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.
bigyikes 5y ago
> In the months before his death in a duel in 1832 at age 20, Galois laid out a new theory of polynomial solutions. That is one hell of a life.

193 points by theafh 5y ago

It's about Scholze work on perfectoid spaces. A nice intro talk in Spanish is at https://cienciastv.org.mx/2020/08/el-mundo-perfectoide-adrian-zenteno/ https://cienciastv.org.mx/2020/08/el-mundo-perfectoide-adria...

infogulch 5y ago
Didn't Kevin Buzzard work on a proof related to perfectoid spaces using Lean theorem prover that featured here a few months ago?
- jesuslop 5y ago
  Yep, you are right. Kevin Buzzard, Johan Commelin, and Patrick Massot, slide 31 at https://robertylewis.com/files/lean_intro_slides.pdf https://robertylewis.com/files/lean_intro_slides.pdf

rohanphadte 5y ago

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.

db48x 5y ago
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.)
- Aerroon 5y ago
  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?
  - darig 5y ago
    Because you didn't manufacture the keyboard you think you want? Enjoy your 3 customers. The rest of us aren't bother to learn the alt codes.
  - yetihehe 5y ago
    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.
  - nabla9 5y ago
    Because Don Knuth only invented digital typesetting he had time to only write TeX and Metafont. Had he also done the same for display managers, we would be fine.
  - db48x 5y ago
    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.

hammock 5y ago

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.

austinheap 5y ago
Yes! I just came here to say this too. For academics: is this tone counter productive? For us lay folk: is this tone helpful?
- scubbo 5y 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.
  - gipp 5y ago
    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.
- abecedarius 5y ago
  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.

8bitsrule 5y ago

This page shows how graphing polynomials can lead to factoring solutions. https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/graphs-of-polynomial-functions/ https://courses.lumenlearning.com/wmopen-collegealgebra/chap...

paulpauper 5y ago

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.

kevinventullo 5y ago
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.
langlands 5y ago
This is actually the first pop math article I've read in a while that didn't make me roll my eyes.

miguelrochefort 5y ago

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.

mycall 5y ago
Graph theories are used for all of your cited examples to explain how they work. They are well studied group homomorphic feature kernels in algebra.
meowkit 5y ago
I believe these are referred to as formal theories and are sort of the root connections between philosophy/epistemology and maths.
langlands 5y ago
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

AndrewKemendo 5y ago

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.

bigyikes 5y ago

> In the months before his death in a duel in 1832 at age 20, Galois laid out a new theory of polynomial solutions. That is one hell of a life.