Speaker:  Ishan Levy (MIT)

Title:  Algebraic K-theory and stable homotopy groups of spheres

Abstract:  The stable homotopy groups of spheres are a mathematical object which play a fundamental role in many areas such as geometric topology. These groups are incredibly complicated and have a rich structure, which is captured via the study of spectra, which are a stable version of the notion of a space. Chromatic homotopy theory provides an approach to studying spectra by decomposing them into simpler `telescopic’ pieces, analogous to the primary decomposition of an abelian group. This decomposition comes from a theorem of Hopkins—Smith giving a rough classification of finite spectra into types. I will explain how algebraic K-theory can refine our understanding of this chromatic filtration. On one hand, the algebraic K-theory of the chromatic filtration captures a refinement of Hopkins—Smith’s classification result, and on the other hand, algebraic K-theory can be used to detect infinitely many linearly independent periodic families in the stable homotopy groups of spheres and disprove Ravenel’s telescope conjecture.