"The more I learned, the more conscious did I become of the fact that I was ridiculous. So that for me my years of hard work at the university seem in the end to have existed for the sole purpose of demonstrating and proving to me, the more deeply engrossed I became in my studies, that I was an utterly absurd person."
Fyodor Dostoevsky, The Dream of a Ridiculous Man
Title: Generalized cycle complexes
Abstract: Given a strictly homotopy invariant sheaf M and smooth scheme X, we define for q≥0 a complex of abelian groups
C*(X, M, q) which combines a Bloch-Levine style cycle complex with coefficients in M with part of the Rost-Schmid complex of M. Using moving lemmas for the Rost-Schmid complex, we show that C*(X, M, q) is quasi-isomorphic to M(q)(X), the q-th layer of the homotopy coniveau tower for M. This allows us to produce a cycle-style complex computing the generalized motivic cohomology groups of X in the sense of Calmès-Fasel, among other applications.
Title: Motivic stable homotopy groups via framed correspondences
Abstract: A modern approach to the motivic stable homotopy category allows one to express its mapping spaces in terms of geometric data called "framed correspondences". We will explain this approach and illustrate it by computing Gm-homotopy groups of the special linear algebraic cobordism spectrum MSL.