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We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a new computational method that yields a streamlined computation of the first 61 stable homotopy groups, and gives new information about the stable homotopy groups in dimensions 62 through 90. The method relies more heavily on machine computations than previous methods, and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.
We prove that the 2-primary $pi_{61}$ is zero. As a consequence, the Kervaire invariant element $theta_5$ is contained in the strictly defined 4-fold Toda bracket $langle 2, theta_4, theta_4, 2rangle$. Our result has a geometric corollary: the 61-s
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.
This paper contains an overview of background from stable homotopy theory used by Freed--Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and
Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $aut_G(S(V)^*)$ denote the space of
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.