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I've been trying lately to understand Fontaine's rings of periods, $B_{\mathrm{dR}}$, $B_{\mathrm{cris}}$, etc. However, I have a really hard time understanding and appreciating how to think about and use these. These rings seems so incredibly complicated and unintuitive it boggles my mindand ; I can never seem to be able to remember their construction.

So, how do I think about, learn and use these obscure objects? (Do I really need to know all the details in their construction to appreciate and use them?)

In addition, I know the context and Fontaine's original motivation for considering these rings, but have they found any unexpected uses outside their intented intended domain?

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# Fontaine's rings of periods

I've been trying lately to understand Fontaine's rings of periods, $B_{\mathrm{dR}}$, $B_{\mathrm{cris}}$, etc. However, I have a really hard time understanding and appreciating how to think and use these. These rings seems so incredibly complicated and unintuitive it boggles my mind and I can never seem to remember their construction.

So, how do I think about, learn and use these obscure objects? (Do I really need to know all the details in their construction to appreciate and use them?)

In addition, I know the context and Fontaine's original motivation for considering these rings, but have they found any unexpected uses outside their intented domain?