The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to have homotopy groups concentrated in even degrees. Despite its extremely simple definition, the even filtration recovers many geometrically and arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the various motivic filtrations on topological Hochschild homology and its variants.
A conjecture of Robert Burklund and Achim Krause predicts that the when applied to l-adic algebraic K-theory of a field, the even filtration should recover the motivic filtration of Voevodsky, relating the concept of evenness to algebraic cycles. In this talk, I will talk about a proof of this conjecture for global and local fields.