Here are seminars I have helped organize, as well as notes accompanying those seminars or other classes.
NACHOS (Non-Abelian Chabauty and Higher Obstructions Seminar): Summer 2017
Schedule: M-W, sometime in the afternoon.
Syllabus: A combination of the following papers:
https://people.maths.ox.ac.uk/kimm/papers/cambridgews.pdf (Kim)
http://www.kurims.kyoto-u.ac.jp/~prims/pdf/45-1/45-1-4.pdf (Kim)
https://arxiv.org/abs/1704.00555 (Brown)
http://hss.ulb.uni-bonn.de/2010/2217/2217.htm (Hadian-Jazi)
A seminar postmortem message to participants, which may be helpful to others in the future:
The real-life technical details of "Kim's cutter," including technical definitions of each map involved (I wish I had known how helpful this paper was earlier) -- for example, what is an admissible de Rham torsor, and how do we know that we get admissible de Rham torsors when all's said and done? For all that and more, there's Kim's Unipotent Albanese and Selmer Varieties for Curves paper.
Even more background material on unipotent groups and the p-adic Hodge theory involved in the constructions: Alex Bett's thesis.
One resource that doesn't fit anywhere specific here is a recent paper of Hast and Ellenberg which does a good job of explaining both the generalities of the theory and the specifics of Faltings' theorem for CM Jacobians in a concise and understandable manner.
Now for some of the more technical ingredients of non-abelian Chabauty, knowledge of which was either assumed or often harder to find online.
A wonderful reference for iterated integrals and Chen's
de Rham Theorem: Hain's AWS Lectures on the topic.
Hall bases and their applications: See section 4 of Hall's original paper. Here's a more extensive review of the subject, though Reutenauer doesn't spell out the connection that Kim needs with the associated graded algebra. Remember, the idea is that we use long exact sequences to reduce the cohomology dimension computation to a computation involving the dimensions of successive quotients of the lower central series. These we can compute using Hall bases.
One of the hardest things about trying to find the resources myself was that the theory has been slowly developing for a long time. Thus, for example, parts of the "Unipotent Albanese" paper generalize and clarify the Siegel's theorem paper. Furthermore, it's very hard to know when reading Deligne's book which concepts are only applicable to the case of mixed Tate motives (basically the projective line minus some number of points) and which ideas are more general. The papers I've referenced above seem to have the most bird's-eye view of the subject, and if one really wants the motivic interpretation one can read Deligne's book or more recent papers of Francis Brown that use recent proofs of motivic conjectures in the mixed Tate case: Here and here.
What are Frobenius invariant paths? Furusho's paper helped me a lot. The canonical (ha) explanation of Frobenius-invariant paths is definitely Besser's paper. Keep in mind that Frobenius-invariant paths are closely related to analytic continuation along Frobenius (as explained in Besser's paper) and that technique is much simpler to understand than isocrystals etc. I recommend starting there to get grounded, and Balakrishnan has a good explanation on page 18 of these slides.
What does the canonical de Rham path mean? Deligne's paper, Section 12.4 for the original construction, and Furusho's paper linked above for an English reference.
We also spent a bunch of time talking about Brown's efforts to get explicit equations for integral points in the mixed Tate case. His papers are mostly self-contained.
Then there are the numerous references for "things to do in cohomology:" Poitou-Tate duality, Pontryagin duality, long exact sequences, etc., but all those references seem easy to find, so I'm not going to put them here.