WS25–26
Seminar Galois theory in algebra and topology, co-organised with Yuenian Zhou. You can find all the information at the GRIPS page and the program also here.
Some comments on the program.

If you plan on organising a seminar inspired by this one, here are the main takeaways we got from running it.

On the maths

  • Talk 3: the author uses \(\mathrm{Gal}(K|k)\) to denote both the inverse limit and the group \(\mathrm{Aut}(K|k)\), leading to confusion. (I would advise the speaker to be especially careful in stating proposition 1.3.5.)
  • Talk 7: many properties of profinite groups are taken for granted by the author. We found a good reference for these missing results in Profinite groups. In particular, we needed section 1.2 of that book. Alternatively, a more categorical approach to the notion of profinite groups can help demystify these statements.
  • Talk 10: one needs the content of talk 12 for the main proof of this talk, so we would suggest to change the talk order compared to our program.
  • Talk 13: this talk deals with coherent modules on a Riemann surface, so it is best to keep it after talks 9-11. We also suggest adding a quick discussion of local systems, which is unfortunately missing from our program. The proof of proposition 2.7.5 omits the construction of the appropriate natural transformations, which are needed to conclude. Moreover, the author defines the tensor product of \(\mathcal{O}\)-modules as the section-wise tensor product. I suggest to state that this, in general, does not spit out a sheaf and it is only due to the remark at the beginning of lemma 2.7.4 that this definition works for this talk.
  • Talk 14: the given definition of tensor products of sheaves of \(\mathcal{O}\)-modules is incorrect in the setting of this talk.

Other

  • The main reference Galois Groups and Fundamental Groups is sometimes quite dry in the proofs, and sometimes a bit sloppy in the notation. At times, it can be tricky to parse even for Master’s students.
  • Talk 5: only introducing the notion of covering spaces is very little material for a 90’ talk. I think it might make sense to spend the remaining time to look at many examples. I would suggest the following examples that are not present in the book: the universal covering of a bouquet of two \(\mathbb{S}^1\)s and maybe a non Galois cover (see for example this stackexchange question).
  • Talk 10: the proofs in this talk proved harder to unravel than the rest. I would suggest it to a more independent student, as many details need to be filled in. In the proof of lemma 3.3.6, I would suggest to highlight why the \(a_i\)s can be extended to the whole of \(X \setminus \phi(S)\), whereas the \(f_i\)s cannot.
  • Talk 12: it could be of interest for the audience to hear about the history and motivation behind the development of sheaf theory. Here are two ideas that might be of interest. The development of sheaf theory goes in the opposite direction compared to the talk, i.e. the original definition of a sheaf is that of a space over a topological space \(X\) (satisfying appropriate conditions, see page 2 of the pdf at this page), and only later was the abstract modern formalism introduced. This is still reflected in the notation, for example in the terms section and restriction. More self-evidently, sheaf theory and sheaf cohomology are tools meant to answer the question of “how do local phenomena globalize?”. Historically, a first instance of this were the Cousin problems (see for example this stackexchange answer).
  • Talk 14: as for talk 10, also the proofs in this talk proved harder to unravel than the rest.

References

Tamás Szamuely
Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2009. ISBN: 9780511627064.
John S. Wilson
Profinite Groups (Oxford, 1998; online edn, Oxford Academic, 31 Oct. 2023), accessed 12 Jan. 2026. ISBN: 9780198500827.
WS24–25
SFB PhD Seminar, co-organised with Debam Biswas and Zhenghang Du.
SS2024
SFB PhD Seminar on (co)homology theories, co-organised with Giacomo Bertizzolo.
WS23–24
HIOB Seminar Perfectoid Spaces and applications, co-organised with Debam Biswas and Niklas Kipp.

Extra

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