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In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory, together with their proofs, in the case of infinite perfect fields. These are our extended notes on these talks.
Over any field of characteristic not 2, we establish a 2-term resolution of the $eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local
We establish a kind of degree zero Freudenthal Gm-suspension theorem in motivic homotopy theory. From this we deduce results about the conservativity of the P^1-stabilization functor. In order to establish these results, we show how to compute cert
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an etale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical
We define the Chow $t$-structure on the $infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{cheartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting
We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree $n$ admits an interpretation in terms of unstable ${mathbb A}^1$-homotopy sheaves of the general