# Fortnightly links (173)

I didn't realize the last fortnightly links were posted in early September! I can explain a few weeks of silence on the blog in December because of a planned surgery, but the rest is solely to blame on some kind of inertia. Anyway, time to get back into semi-regular blogging!

Daniel Huybrechts, Dominique Mattei: Splitting unramified Brauer classes by abelian torsors and the period-index problem does something absolutely amazing. The

*period-index conjecture*is an important open problem, trivial in dimension 1, solved through hard work of de Jong and Lieblich in dimension 2, and wide open in higher dimension (except for abelian 3-folds, by amazing work of Hotchkiss and Perry). In this paper, it is shown that for every unramified Brauer class in the function field (which means it comes from an Azumaya algebra on some smooth and proper model) the index divides some power of the period. This doesn't settle the period-index conjecture though, because the power is not given in terms of the dimension, but rather in terms of the geometry of curves on the variety.Vladimiro Benedetti, Daniele Faenzi, Maxim Smirnov: Derived category of the spinor 15-fold checks the Kuznetsov–Smirnov conjecture which relates the finer structure of small quantum cohomology to the existence of full exceptional Lefschetz collections for the spinor 15-fold. Cool! Next up, constructing its homological projective dual? There are various open questions on the Freudenthal variety at the end, in case you are looking for some hard problems to work on!

Nicolas Addington, Elden Elmanto: The Quillen-Lichtenbaum dimension of complex varieties introduces new derived invariants, and it is extremely high on my must-read list, but I haven't managed to do so yet. Its appearance in these fortnightly links is thus also a reminder to myself.

Elliot Kienzle's blog is something I found through his amazing Twitter feed which features the best drawings in mathematics I have ever seen. Highly recommend to follow them if you are still on Twitter.

Tom Coates, Alexander M. Kasprzyk, Sara Veneziale: Machine learning the dimension of a Fano variety is about determining the dimension of a Fano variety from its quantum period, a "fingerprint" of a deformation class of a Fano variety which is defined in terms of its Gromov–Witten theory (thus its symplectic geometry). If it is to be a good fingerprint (in the sense that it determines the deformation class) it should certainly contain enough information to determine the dimension of the Fano variety. The paper does not give a proof for this, but using machine learning methods, they realize that computers can be trained to predict the dimension pretty well. Super-duper cool!