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Topology Seminar

Thu., Mar. 10, 2022 1:00 p.m.

Location: Live Stream

Speaker: William Balderrama, University of Virginia

Title: The motivic lambda algebra and Hopf invariant one problem



Current best approaches to understanding the stable homotopy groups of spheres at the prime 2 make use of the Adams spectral sequence, which computes stable stems starting with information about the cohomology of the Steenrod algebra. The first major success of the Adams spectral sequence was in Adams' resolution of the Hopf invariant one problem, which proceeded via an analysis of secondary cohomology operations. Later, J.S.P. Wang used a certain algebraic device, the lambda algebra, to give a more thorough computation of the cohomology of the Steenrod algebra, and used this to give a slick almost entirely algebraic derivation of the Hopf invariant one theorem.

In this talk, I will go over some of the above history, and then describe work (joint with Dominic Culver and J.D. Quigley) on analogues in motivic stable homotopy theory. In particular, I will describe a mod 2 motivic lambda algebra, defined over any base field of characteristic not equal to 2, as well as some of what can be said about the 1-line of the motivic Adams spectral sequence for various base fields.