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Stream: deprecated: thermodynamics

Topic: exterior differential systems

Owen Lynch (Dec 04 2021 at 16:50):

Have any of you all come across exterior differential systems before? https://link.springer.com/book/10.1007/978-1-4613-9714-4

Seems like a very interesting generalization of ordinary and partial differential equations.

Evan Patterson (Dec 04 2021 at 21:10):

I stumbled across this stuff when learning about exterior calculus but never dug into it. Looks interesting though.

Owen Lynch (Dec 04 2021 at 21:12):

That makes sense that it came up w.r.t. exterior calculus.

I particularly like the condition that the ideal be closed under differentiation, because it makes a lot of sense. I.e., if you have some relation $X=Y$, then you definitely also want $\dot{X} = \dot{Y}$.

Owen Lynch (Dec 04 2021 at 21:17):

Actually, just straight away I think we have a category here that is interesting. The objects are manifolds, and a morphism between the spaces $X$ and $Y$ is an ideal of $\Omega^\ast(X \times Y)$. Given an ideal $I \subset \Omega^\ast(X \times Y)$ and an ideal $J \subset \Omega^\ast(Y \times Z)$, I think we can make an ideal of $\Omega^\ast(X \times Z)$ by taking... $I+J$ as an ideal of $\Omega^\ast(X \times Y \times Z)$ and then projecting down?? Something like that.

Owen Lynch (Dec 04 2021 at 21:19):

I need to work out an example.

Owen Lynch (Dec 04 2021 at 21:24):

If $I$ is generated by $dx - dy$, and $J$ is generated by $dy - dz$, then $J \circ I$ should be generated by $dx - dz$. Which is something that is in $I+J$. I think instead of projecting down, we take the inverse image from the map $\Omega^\ast(X \times Z) \to \Omega^\ast(X \times Y \times Z)$. There's some double contravariance going on here: the natural map $X \times Y \times Z \to X \times Z$ induces a map $\Omega^\ast(X \times Z) \to \Omega^\ast(X \times Y \times Z)$, which in turn induces a map from ideals of $\Omega^\ast(X \times Y \times Z)$ to ideals of $\Omega^\ast(X \times Z)$

Owen Lynch (Dec 04 2021 at 21:25):

OK, and I'm pretty sure this is a symmetric monoidal category! The monoidal product is the cartesian product of manifolds (which is not the categorical product in this category)

Owen Lynch (Dec 04 2021 at 21:25):

Neato!

Owen Lynch (Dec 04 2021 at 21:26):

Hey, if we use the graded ring of discrete exterior forms, this might even be useful to GATEM!

Evan Patterson (Dec 04 2021 at 21:28):

Nice! Yeah, it would be cool if we could compose up a meaningful PDE using this category.

Evan Patterson (Dec 04 2021 at 21:29):

Is the category really cartesian though? It seems more relational in character to me.

Owen Lynch (Dec 04 2021 at 21:29):

Oh, you are totally right

Owen Lynch (Dec 04 2021 at 21:29):

It's monoidal using the cartesian product of manifolds, but that's definitely not the categorical product in this category

Owen Lynch (Dec 04 2021 at 21:32):

Evan Patterson said:

Nice! Yeah, it would be cool if we could compose up a meaningful PDE using this category.

Hmmmm, one meaningful PDE coming up!

Owen Lynch (Dec 04 2021 at 21:47):

OK, first of all, I think I messed up with the definition. I think the morphisms should be ideals of $\Omega^\ast(X) \oplus \Omega^\ast(Y)$, where that $\oplus$ is the direct sum of graded rings. This is important, because there are morphisms $A \oplus B \to A$, and morphisms $A \to A \oplus B$, which means that we can transfer ideals in both directions, which is necessary for composition.

Owen Lynch (Dec 04 2021 at 21:54):

OK, I have some harebrained example for what this means, but I think it's too harebrained to inflict on the world for now...

Owen Lynch (Dec 04 2021 at 21:54):

Gotta actually work out some math before I confuse everything with a bunch of false statements

John Baez (Dec 04 2021 at 22:34):

Some of this stuff can be seen as algebra: that is, the study of dgas, meaning differential graded algebras. Differential forms on a manifold form a dga, and there's a sort of obvious concept of 'ideal' for dgas, which may be called a 'differential graded ideal': you can mod out a dga by a differential graded ideal and get a dga.

John Baez (Dec 04 2021 at 22:36):

There's a huge amount of math about dgas, only a tiny bit of which can be found on the link. If you're mainly interested in differential forms, then you want 'commutative' dgas, where

$\mu \wedge \nu = (-1)^{pq} \nu \wedge \mu$

where $\mu$ has grade $p$ and $\nu$ has grade $q$. These act a lot like commutative rings: e.g. you don't need to worry about the different between left, right and two-sided ideals.

Paolo Perrone (Dec 06 2021 at 10:46):

Ideals being closed under differentiation seems to be the right notion for a lot of things. From the geometrical perspective, they are linked to involutive distributions, that is, the one that are (thanks to Frobenius' theorem) integrable.