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Can anybody recommend a good reference to learn the Levelt-Turritin decomposition theorem of formal connections? An intuitive description of what it says would also be very appreciated.

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For a new approach and intuitive explanation see this article. In the older literature, Levelt's Original paper is the best reference I have found! In particular, it is more understandable than many references that come after. The statement is very simple: every differential operator over the field $\overline{\mathbb{C}((t))}$ has a Jordan canonical form.

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