…which are admittedly lacking in this disaster of a year. Nonetheless, here they are (in no particular order – note that these are not theorems proven this year, only that I read about them this year)
- (Kerz-Strunk-Tamme)Let be a -dimensional Noetherian scheme. Then vanishes for .
One of the longest standing conjectures in algebraic -theory is finally resolved! The proof involves the invention of derived blowups, which makes it all the better.
- (Blumberg-Gepner-Tabuada) There is a stable -category of noncommutative motives such that (nonconnective) algebraic -theory is corepresentable by the “sphere spectrum.”
There is a poster which states something like “math is 1% geometry, 0.1% noncommutative geometry and 98.9% dark geometry (categories).” Nothing captures that spirit more than this fantastic theorem, that pushes you down the inspiring rabbit hole of higher category theory.
- (Antieau-Gepner-Heller) Various theorems on negative -theories, including a nonconnective theorem of the heart.
Negative -theory somehow popped up a lot this year for me, one big reason is probably this paper which pushes our frontiers down the connective covers.
- (Lieblich, De Jong) Let be a smooth, projective, connected surface over an algebraically closed field . Let be a Brauer class, then .
A baby wields numbers, a teenager wields abelian groups and perhaps adults wield Brauer classes – that is – we can now “geometrize” divisibility relations.
- (Gaitsgory-Lysenko) The start of the metaplectic Langlands program.
It’s Langlands, with a twist – now with Brauer groups and !
Who put the Steenrod squares in my analytic cycles? Well Atiyah-Hirzerbruch first did, but these recent, topological additions to the counterexamples continues the trend of topologists trolling the sacred grounds of geometry.
- (Wickelgren-Kass) There is a theory of -Brouwer degree, which interprets the EKL class, and is useful to classify local singularities of varieties over arbitrary fields.
Last, but not least, are three theorems – two of which belong to Robert Thomason and the last one inspired by his work. This year, his notebooks were archived and they are a delightful read as one would expect from one of the masters.
- (Thomason) Full dense triangulated subcategories of a triangulated category is classified by subgroups of .
Seems to not be one of his famous theorems, but one (of three) of his last. A quite remarkable observation on how much even sees about the whole structure of categories.
- (Thomason) Bott-inverted algebraic -theory satisfies étale descent. (I learned from Joe Berner that this abbreviated as
AKétAKTEC by people who know what they are talking about)
I think enough have been said about this classic, and its applications, which leads me to:
- (Clausen, Mathew, Noel, Naumann) Let be a map of -rings such that the induced map is surjective on rational , then various theories satisfy descent for after periodic localization.
which explains in modern terms what went on in 1984 (and so much more!). The idea is elegant – one uses power operations (manifested in the form of May’s nilpotence conjecture) to spread rational information along the periodic localizations and the execution is uncompromisingly modern!