3 ms·

https://www.youtube.com/watch?v=4dyytPboqvE https://www.youtube.com/watch?v=4dyytPboqvE

munchler 2y ago
Thank you. Watching it now. Does the Langlands program compete with Category Theory as being the grand unified theory of math, or are they not really comparable?
- moi2388 2y ago
  Not a mathematician. Afaik category theory is more like an alternative to set theory. Langlands is more like a bridge between higher level mathematics, allowing you to transform hard problems in geometry to harmonic analysis and vice versa, and so far specifically these fields only.
  - auntienomen 2y ago
    Yeah, category theory is a framework for describing mathematical structure. It's not vacuous -- there are mathematical structures which don't fit into the framework and there are some theorems about what properties a category has. But category theory doesn't do that much on its own. The geometric Langlands conjectures are a _lot_ more specific, and a lot more focused. They're a big deal, because they're a toy model for the arithmetic Langlands conjectures, which are a generalization of the machinery that proved Fermat's Last Theorem and would give an effective method for dealing with a lot of number theory problems.
- hackandthink 2y ago
  I do not know anything about the Langlands program. But I see a lot of categories and functors, so I guess they use, speak and think Category Theory.
- moomin 2y ago
  It’s actually much more interesting. Category Theory is one of a number of systems that can underpin regular mathematics. The root of the tree, if you will. There exist proofs that these systems are equivalent and they’re not that hard to follow. But there’s a lot of branches of the tree of mathematics involving extremely different constructions. What is proven here is that two branches of mathematics have a logical equivalence. Between this and other work, it’s looking increasingly like large numbers of the branches of the tree are effectively the same. This is insanely hard to understand right now, but hopefully in the future this will lead to a whole new understanding of mathematics where these correspondences are natural.
nextaccountic 2y ago
This is about the Langlands program, not geometric Langlands

Thank you. Watching it now. Does the Langlands program compete with Category Theory as being the grand unified theory of math, or are they not really comparable?

moi2388 2y ago
Not a mathematician. Afaik category theory is more like an alternative to set theory. Langlands is more like a bridge between higher level mathematics, allowing you to transform hard problems in geometry to harmonic analysis and vice versa, and so far specifically these fields only.
- auntienomen 2y ago
  Yeah, category theory is a framework for describing mathematical structure. It's not vacuous -- there are mathematical structures which don't fit into the framework and there are some theorems about what properties a category has. But category theory doesn't do that much on its own. The geometric Langlands conjectures are a _lot_ more specific, and a lot more focused. They're a big deal, because they're a toy model for the arithmetic Langlands conjectures, which are a generalization of the machinery that proved Fermat's Last Theorem and would give an effective method for dealing with a lot of number theory problems.
hackandthink 2y ago
I do not know anything about the Langlands program. But I see a lot of categories and functors, so I guess they use, speak and think Category Theory.
moomin 2y ago
It’s actually much more interesting. Category Theory is one of a number of systems that can underpin regular mathematics. The root of the tree, if you will. There exist proofs that these systems are equivalent and they’re not that hard to follow. But there’s a lot of branches of the tree of mathematics involving extremely different constructions. What is proven here is that two branches of mathematics have a logical equivalence. Between this and other work, it’s looking increasingly like large numbers of the branches of the tree are effectively the same. This is insanely hard to understand right now, but hopefully in the future this will lead to a whole new understanding of mathematics where these correspondences are natural.

nextaccountic 2y ago

This is about the Langlands program, not geometric Langlands