Gems of 2016

…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 $X$ be a $d$ -dimensional Noetherian scheme. Then $K_i(X)$ vanishes for $i< -d$ .

One of the longest standing conjectures in algebraic $K$ -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 $\infty$ -category of noncommutative motives such that (nonconnective) algebraic $K$ -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 $K$ -theories, including a nonconnective theorem of the heart.

Negative $K$ -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 $X$ be a smooth, projective, connected surface over an algebraically closed field $k$ . Let $\alpha$ be a Brauer class, then $per(\alpha) = ind(\alpha)$ .

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 $K_2$ !

(Antieau, Kameko, Tripathy) Counterexamples to the integral Hodge conjecture.

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 $\mathbb{A}^1$ -Brouwer degree, which interprets the EKL class, and is useful to classify local singularities of varieties over arbitrary fields.

Throwback to the times when Brouwer was stirring his coffee, when Milnor was fibering spheres over circle, and when every sphere was singularly exotic. Then add Morel and Voevodsky.

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 $T$ is classified by subgroups of $K_0(T)$ .

Seems to not be one of his famous theorems, but one (of three) of his last. A quite remarkable observation on how much even $K_0$ sees about the whole structure of categories.

(Thomason) Bott-inverted algebraic $K$ -theory satisfies étale descent. (I learned from Joe Berner that this abbreviated as ~~AKét~~ AKTEC 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 $f: A \rightarrow B$ be a map of $E_{\infty}$ -rings such that the induced map is surjective on rational $K_0$ , then various theories satisfy descent for $f$ 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!

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