Kervaire Invariant One [after M. A. Hill, M. J. Hopkins, and D. C. Ravenel]

The question of when the Kervaire invariant is nontrivial was the only question left unresolved by Kervaire and Milnor in their 1963 study of the relationship between groups of homotopy spheres and stable homotopy groups. In 2009, Mike Hill, Mike Hopkins, and Doug Ravenel resolved this question except in one dimension, by a highly innovative attack using large amounts of equivariant stable homotopy theory and small amounts of computation. The present paper is a Seminaire Bourbaki report on this work.Comment: This is a the submitted Seminaire Bourbaki report. 30 page

On the anti-automorphism of the Steenrod algebra: II

The relations of Barratt and Miller are shown to include all relations among the elements $P^i\chi P^{n-i}$ in the mod $p$ Steenrod algebra, and a minimal set of relations is given.Comment: 6 page

The Burnside bicategory of groupoids

Several models for the Burnside bicategory of groupoids are described and shown to be equivalent. As observed by the late Gaunce Lewis, the corresponding Burnside category is additive.Comment: 21 pp. This draft improves the paper following referee's suggestion

Language, Technology, and Engagement in the Haitian Classroom: An Interim Report on the MIT-Haiti Initiative

Since 2013 I have been traveling to Haiti as part of the MIT- Haiti Initiative. This initiative, led by Professor Michel DeGraff of the MIT Department of Linguistics and Philosophy, aims to encourage active learning strategies, enabled by technology when possible and appropriate, and strongly stresses the importance of the use of the one language spoken by all Haitians, namely Haitian Creole (or Kreyòl). To use a Haitian metaphor, these three components form the three stones on which the cook-pot of our educational approach rests. We have focused our attention on the higher education sector. In this note I will begin with a review of the educational landscape that forms the background for our efforts. Then I will describe the work of the MIT-Haiti Initiative and some of the efforts undertaken in Haiti by participants in our workshops. I will discuss the mathematical material developed for these workshops and what we learned in leading them, and then describe the findings of a site visit to a campus of the State University of the Haiti. I will end by discussing how the typography of equity spelled out by Rochelle Gutiérrez (Teaching for Excellence and Equity in Mathematics, 2009) applies in the Haitian educational setting

Inverting the Hopf map

We calculate the η-localization of the motivic stable homotopy ring over C, confirming a conjecture of Guillou and Isaksen. Our approach is via the motivic Adams-Novikov spectral sequence. In fact, work of Hu, Kriz and Ormsby implies that it suffices to compute the corresponding localization of the classical Adams-Novikov E₂-term, and this is what we do. Guillou and Isaksen also propose a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov

On a spectral sequence for the cohomology of infinite loop spaces

We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a connective spectrum X. Under good conditions its E₂-term is computable as certain nonabelian derived functors evaluated at H* (X) as a module over the Steenrod algebra, and it converges to the cohomology of Ω ∞ X. We provide general methods for computing the E₂-term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at E₂ when X is a suspension spectrum

Harrison homology and the Quillen homology of commutative monoids

The cohomology theory for commutative monoids developed by P. A. Grillet is a case of a graded form of Harrison homology