In my PhD thesis I gave a bound for the Gelfand-Kirillov dimension of admissible $p$-adic Banach representations with an infinitesimal character for $\mathrm{GL}_2$ of a $p$-adic field. I also have a bound for families: the article is being written, but you can already find it in my thesis (below).
I am currently interested in extending this beyond $\mathrm{GL}_2$ and in applications of the bound, to completed cohomology for example.
Preprints & publications
Version 2 almost finished (fixing many minor errors + clarifying main argument)
[arXiv]
manuscripta mathematica
[journal] [arXiv]
Theses
My three theses. Supervised by Arno Kret and Benjamin Schraen, Benjamin Schraen, and Arno Kret, respectively.PhD thesis:
Gelfand-Kirillov dimensions and the $p$-adic Langlands programAccessible at UvA-DARE and soon, after some minor corrections, on the database of Université Paris-Saclay.
For the moment, this is the only place where the following results can be found:
- Gelfand-Kirillov bound for $\mathrm{GL}_2$ in the families case (Chapter 5). Writing a preprint about this is a high priority for me. Mainly, I plan to add some more applications.
- Computation of classical Hecke action on derived pro-$p$ Iwahori invariants (Chapter 6). This is done in the setting of Breuil-Herzig-Hu-Morra-Schraen, but can probably be done more generally using the family result refered to above. More general or not, I should make a preprint out of it some time, but this is not a top priority for me.
Mémoire de M2 (expository):
Lubin-Tate towers [pdf]Very concrete and algebraic (not so geometric) approach to Lubin-Tate towers, via formal groups following Gross-Hopkins. Constructing the higher level towers à la Drinfeld and (almost) showing perfectoidness at infinite level following Weinstein and Scholze-Weinstein.
The writing process was quite painful and it took my whole Summer to complete :( Afterwards I quickly tried to forget it. There are probably many mistakes as well and you might be the first to read it!
A week after submitting it, I came up with a proof for the unproven claim at the end (resulting in a +1 to my note!).
Undergraduate thesis (expository):
Chebotarev's Density Theorem [pdf]The idea of the project was to give the "first detailed modern exposition" of Chebotarev's density theorem without using class field theory. I actually doubt it was the first, but I did give a proof.
I mostly followed a book of Lang. Then I avoided class field theory following an exposition of Lenstra and Stevenhagen (of the original strategy of Chebotarev himself, who of course also 'avoided' class field theory).
Later on, my advisor told me he found some unspecified mistakes while using my thesis in a course. One should ask him about the mistake or (better) ask me to ask him. I only found some very minor but very embarrasing mistakes immediately after submission!