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I am a graduate student with some background in Galois deformation theory. I am familiar with the basics (the existence of a universal deformation space with prescribed conditions) and with some examples in Galois deformation theory, as well as with the some of the conjectural relations with $p$-adic systems of Hecke eigenvalues (i.e. $R=T$ theorems). I have seen applications and arguments in the case of $\operatorname{GL}_2$.

I'd like to get myself to a spot where I am somewhat familiar with the more modern advancements in the theory, by which I mean, Taylor–Wiles–Kisin patching, Kisin's deformation rings, potential modularity theorems, the Calegari–Geraghty method, the 6-author "Patching and the $p$-adic local Langlands correspondence" paper, etc...

It seems not so easy to start reading such papers without an idea where you're going because frequently they approach 100 pages with many logical dependencies. On the other hand, the more introductory accounts like the Darmon-Diamond-Taylor papers appears to be slightly outdated (as far as I understand).

So, what would be a good roadmap for studying Galois def theory/modularity theorems from a modern perspective? Where should I begin?

Thanks!

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A fantastic place to start would be Toby Gee's notes from the 2013 Arizona Winter School. This gives a nice overview of the theory as it then existed -- things have of course moved on further since then, but it's significantly more "modern" than Darmon--Diamond--Taylor, for instance. The course concludes with a proof of a GL(2) modularity lifting theorem over totally real fields, a result which goes back to Fujiwara in 2000 or so, but rather than presenting Fujiwara's proof, the notes give a "modernized" proof using more up-to-date machinery.

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  • $\begingroup$ Dear David, thank you for your answer. I've read most of these notes and did indeed find them useful. Do you have an opinion on what could be a second (more advanced) source to look at after these notes? For example, what could one read in order to understand better the geometry of the various deformation spaces/Hecke algebras? $\endgroup$
    – xlord
    Sep 13, 2020 at 9:32
  • $\begingroup$ Once you've got your head around Toby's notes, you should be in a good position to start reading the research literature. The notes include some suggestions for further reading, which might give you some good pointers. (Your request for a "second thing to read" suggests that you believe the subject to be more "linear" in its development than it actually is -- like any subject with >1 person working on it, there isn't a unique correct order to read things. There are many roads to modularity lifting, none of which is the royal road.) $\endgroup$
    – David Loeffler
    Sep 14, 2020 at 9:38

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