Aleksander Shmakov

Postdoc at Stockholm University. Arithmetic geometry and adjacent topics.

[Edgar Shaghoulian @eshaghoulian]

https://t.co/MuMgadqGeK

[Aleksander Shmakov @AShmakovMath]

@dzackgarza (cutting out paper models for elementary school students to tape together) okay kids today we are learning about K3 surfaces

[Aleksander Shmakov @AShmakovMath]

@dzackgarza Here's some nice results in the case of a K3 times an elliptic curve: https://t.co/9EvKFlWhOp

[Aleksander Shmakov @AShmakovMath]

@dzackgarza Me forgetting all the real analysis I know only to end up in Langlands (famously known for its lack of L^2 spaces)

[Aleksander Shmakov @AShmakovMath]

@dzackgarza It's the (2,3,\infty) triangle group.

[Aleksander Shmakov @AShmakovMath]

@dzackgarza Morel's IHES summer school lectures on Shimura varieties helped me a lot with Shimura data. Maybe you will find them similarly helpful, if you have not seen them already! https://t.co/5kDuBwOSAE

[Aleksander Shmakov @AShmakovMath]

@d_m_d_m_d_d It *can* be written as, but *should* it be written as?

[Aleksander Shmakov @AShmakovMath]

@dzackgarza Calling SL_3 type G2 (simply laced version)

[Aleksander Shmakov @AShmakovMath]

@anton_hilado Every one knows (or should know) what it is:
(object that can be written in at least 5 different ways)

[Aleksander Shmakov @AShmakovMath]

@anton_hilado Oh also, if we restrict to the everywhere unramified situation then there is a classification up to motivic weight 22 by Chenevier and Lannes which massively narrows down the motives which can appear in this range. See this paper of Bergström and Faber: https://t.co/l7RqPLOvxS

[Aleksander Shmakov @AShmakovMath]

@anton_hilado I've heard that the motive of a K3 surface always embeds into the motive of an Abelian variety, possibly of rather high dimension. But then for Abelian varieties, the story of potential modularity for anything beyond Abelian surfaces or 3-folds gets quite unclear!

[Aleksander Shmakov @AShmakovMath]

@anton_hilado There are some nice results for singular K3 surfaces over Q being modular (related to cusp forms of weight 3 for \Gamma_0(N) with CM), and similarly for rigid Calabi-Yau 3-folds over Q (related to cusp forms of weight 4 for \Gamma_0(N)). But in general this is hard!

[Aleksander Shmakov @AShmakovMath]

@littmath @kittegory The powers of the canonical bundle are hidden in the higher homotopy groups of this sheaf of spectra! Their cohomology shows up in the descent spectral sequence computing the homotopy groups of Tmf, so some part of these homotopy groups sees modular forms.

[Aleksander Shmakov @AShmakovMath]

@dzackgarza What book is this from? This looks lovely!

[Aleksander Shmakov @AShmakovMath]

@dzackgarza I have something like this that is a block away from my apartment and gives you a free coffee order every 2 hours for $20 a month and I am going to miss it when I leave Toronto

[Aleksander Shmakov @AShmakovMath]

@littmath "In my more advanced moments I remember all the time that I have no control... There is always going to be a part of every single build in which I am so mystified I wonder what the hell I'm even doing there. It really is just part of the process." https://t.co/RlnxR8lKVu

[Aleksander Shmakov @AShmakovMath]

@anton_hilado lol at this being immediately followed by "Recall that"

[Aleksander Shmakov @AShmakovMath]

@dzackgarza People understand quantum gravity, just in dimensions lower than what is physically relevant. As an example that is maybe relevant to what you are studying, Witten's conjecture is about quantum gravity in dimension 2, and involves intersection theory on moduli of curves.