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What is the difference between Langlands program and Category theory?

pmiller2 7y ago
You may find this helpful: https://en.wikipedia.org/wiki/Langlands_program https://en.wikipedia.org/wiki/Langlands_program
tprice7 7y ago
They are different enough that it's frankly a bit difficult to make sense of that question. I think in general, when asking "what is the difference between X and Y" questions, it helps if you elaborate a bit on why you think X and Y are similar / difficult to distinguish. That gives the person who answers the question a bit of a starting point: they can talk how the similarities you perceive might not be the whole story, or might be illusory (and you might end up answering your own question while clarifying your thoughts on the matter). Otherwise it might not be clear how to even begin answering the question.
- tprice7 7y ago
  Reflecting on this more, my best guess is that you think they are similar because they are both a "grand unified theory of mathematics". To explain the difference extremely briefly then, I'd say it's a massive exaggeration to call either category theory or the Langlands program a "grand unified theory of mathematics." They each demonstrate connections between some areas of mathematics, but they are different connections, and fall very short of unifying everything. If you want to know more, I think you'd be better off learning more about category theory and the Langlands program independently of each other rather than looking for a comparison between them.
auntienomen 7y ago
It's the difference between a typed language and a program written in that language.
mikorym 7y ago
The two are not related in terms of people working in the fields. I have never heard anyone from my category theory circles mention the Langlands program and saw it for the first time on Wikipedia. However, skimming the Wikipedia page, I can see a few interesting things, e.g. Galois groups and a notion of functoriality [1] that are analogous to concepts in category theory. It doesn't look like the Langlands program has been "Categorified" a la Grothendieck yet. [1] https://en.wikipedia.org/wiki/Langlands_program#Functoriality https://en.wikipedia.org/wiki/Langlands_program#Functorialit...

You may find this helpful: https://en.wikipedia.org/wiki/Langlands_program https://en.wikipedia.org/wiki/Langlands_program

tprice7 7y ago

They are different enough that it's frankly a bit difficult to make sense of that question. I think in general, when asking "what is the difference between X and Y" questions, it helps if you elaborate a bit on why you think X and Y are similar / difficult to distinguish. That gives the person who answers the question a bit of a starting point: they can talk how the similarities you perceive might not be the whole story, or might be illusory (and you might end up answering your own question while clarifying your thoughts on the matter). Otherwise it might not be clear how to even begin answering the question.

tprice7 7y ago
Reflecting on this more, my best guess is that you think they are similar because they are both a "grand unified theory of mathematics". To explain the difference extremely briefly then, I'd say it's a massive exaggeration to call either category theory or the Langlands program a "grand unified theory of mathematics." They each demonstrate connections between some areas of mathematics, but they are different connections, and fall very short of unifying everything. If you want to know more, I think you'd be better off learning more about category theory and the Langlands program independently of each other rather than looking for a comparison between them.

auntienomen 7y ago

It's the difference between a typed language and a program written in that language.

mikorym 7y ago

The two are not related in terms of people working in the fields. I have never heard anyone from my category theory circles mention the Langlands program and saw it for the first time on Wikipedia. However, skimming the Wikipedia page, I can see a few interesting things, e.g. Galois groups and a notion of functoriality [1] that are analogous to concepts in category theory. It doesn't look like the Langlands program has been "Categorified" a la Grothendieck yet. [1] https://en.wikipedia.org/wiki/Langlands_program#Functoriality https://en.wikipedia.org/wiki/Langlands_program#Functorialit...