Title: What happens when you let a topologist play with cycles
Abstract: I will explain one of two (or both!) "results" on (algebraic) cycles.
1) I first learned from Hélène a result of Bloch, using Voevodsky's resolution of the Bloch-Kato conjecture, that any p-torsion integral topological cycle on a smooth projective C-variety vanishes off a codimension 1 subvariety. There is a generalization of this where one can let 1 be a bigger number and one can replace p-torsion with "v_n"-torsion in the sense of algebraic topology. This is joint work with Robert Burklund.
2) One can formulate the crystalline version of the integral Tate "conjecture." This is, of course, expected to be false. Still, it fails for an interesting reason which I will explain using a version of "genuine" Steenrod operations on crystalline/de Rham cohomology and how to port Atiyah-Hirzebruch's (probably the first time topologists played with cycles) counterexamples to this setting. This is joint work with Arpon Raksit.