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)

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.

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.

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.

A baby wields numbers, a teenager wields abelian groups and perhaps adults wield Brauer classes – that is – we can now “geometrize” divisibility relations.

It’s Langlands, with a twist – now with Brauer groups and $K_2$!

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.

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.

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.

I think enough have been said about this classic, and its applications, which leads me to:

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!