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Linear ordinary differential equations is a topic that you might have first encountered in a calculus course, but they are secretly hiding topological structure! This idea goes back to Riemann, and is the subject of one of Hilbert’s famous questions. 1/n
Linear ordinary differential equations are defined over the real line, or the complex plane. But they can be generalised to gadgets living over any smooth manifold. These generalised differential equations are called flat connections. 2/n
The famous Riemann-Hilbert correspondence is an equivalence between the category of flat connections on a manifold M, and the category of representations of its fundamental group. 3/n
Stating this as an equivalence of categories is just a fancy way of saying that there is a well-defined recipe for going back and forth. This recipe is differentiation and integration! 4/n
The RHC is a special case of a more general equivalence. Another special case is the equivalence between the category of representations of a simply connected Lie group, and the category of representations of its Lie algebra. This follows from Lie’s second theorem. 5/n
In order to unify both pictures, I need to tell you a bit about Lie groupoids and Lie algebroids. 6/n
Lie groupoids are just like Lie groups, but they have more than one identity element. They also have a multiplication, but it is only partially defined, for 'arrows' whose endpoints match up. 7/n
The archetypal example of a Lie groupoid is the fundamental groupoid of a manifold M. It consists of (homotopy classes of) paths in M, and the multiplication is given by sticking one path after the other. 8/n
Lie algebroids generalize Lie algebras. It’s useful to think of them as gadgets that mimic the tangent bundle of a manifold. The tangent bundle is the archetypal example, but we could replace it with another Lie algebroid, and we might not even notice the difference. 9/n
Many of the amazing results about Lie groups generalize to Lie groupoids. For example, a Lie groupoid determines a Lie algebroid. And a version of Lie's second theorem holds as well. 10/n
https://ncatlab.org/nlab/show/Lie%27s+three+theorems#generalization_of_lies_theorems_to_lie_groupoids
https://ncatlab.org/nlab/show/Lie%27s+three+theorems#generalization_of_lies_theorems_to_lie_groupoids
Lie’s second theorem implies that under certain connectivity assumptions, there is an equivalence between the category of representations of a Lie groupoid, and the category of representations of its Lie algebroid. 11/n
Why is this is relevant to differential equations? The Lie algebroid of the fundamental groupoid is the tangent bundle. And representations of the tangent bundle are the same thing as flat connections! 12/n
Lie’s second theorem therefore get’s us part ways to the Riemann-Hilbert correspondence: it gives us an equivalence between flat connections on M and representations of the fundamental groupoid of M. 13/n
A representation of the fundamental groupoid is just what you think it is: a linear isomorphism for every path in M, all chosen in such a way that composition of paths is respected. 14/n
To get all the way to the Riemann-Hilbert correspondence, we need one more ingredient: Morita equivalence. 15/n
Roughly, two groupoids are Morita equivalent if they determine the same quotient space M/G. This can be made precise using the language of differentiable stacks. 16/n
Morita equivalent Lie groupoids have equivalent categories of representations. This is because the representations give the vector bundles over the quotient space (stack). 17/n
The fundamental groupoid of a manifold is Morita equivalent to the fundamental group at any of its basepoints, and therefore they have equivalent categories of representations. This completes the Riemann-Hilbert correspondence. 18/n
In summary, the classical Riemann-Hilbert correspondence can be viewed as a consequence of two very general types of equivalence: Morita equivalence, and Lie's second theorem. 19/n
Some of these same ideas can be applied to study differential equations with singularities, a subject which also goes back to work of Riemann and Fuchs, among others. I may get to this later. 20/n
Here are some references:
* Lie Groupoids: https://arxiv.org/abs/math/9602220
* Morita equivalence: https://arxiv.org/abs/math/0203100 (section 2)
* Groupoids and singular differential equations: https://arxiv.org/abs/1305.7288
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* Lie Groupoids: https://arxiv.org/abs/math/9602220
* Morita equivalence: https://arxiv.org/abs/math/0203100 (section 2)
* Groupoids and singular differential equations: https://arxiv.org/abs/1305.7288
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