"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: From strictly homotopy invariant sheaves to
effective homotopy modules
Abstract: Let SHS1(k) and SH(k) denote the motivic S1- and P1-stable homotopy categories, respectively. They are related by the
Gm-stabilization adjunction SHS1(k) <-> SH(k). Following Levine-Voevodsky, this adjunction can be factored further as
SHS1(k) <-> SHS1(k)(1) <-> ... <-> SH(k).
Each of the above categories has a t-structure, and all of the adjunctions are compatible with the t-structures
(i.e. the left adjoints are right-t-exact). In studying the effect of Gm-stabilization on the hearts, we are lead to the following
Conjecture: the functor SH(k)♥ -> SHS1(k)(n)♥ is an equivalence for n > 0.
Using the theory of framed transfers and the homotopy coniveau tower, we provide two pieces of evidence for this conjecture:
(1) the functor is fully faithful, and (2) if M is a strictly homotopy invariant sheaf, then M-2 is an effective homotopy module.
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.