Stable homotopy groups, Higher algebra and the Telescope Conjecture

A fundamental motivating problem in homotopy theory is the study of the stable homotopy groups of spheres. The mathematical object that binds stable homotopy groups together is a spectrum. In this talk we will adopt the viewpoint that spectra are to be seen as the homotopy theoretic counterparts of abelian groups. Just as abelian groups form the foundational pillar for algebra and algebraic geometry, one can  develop  “Higher Algebra ” where spectra play a comparable role. Via this perspective the study of spectra is done by a local to global approach. Where spectra are decomposed into so-called “monochormatic layers”.  I shall describe recent advancements in the study of these monochromatic layers including the disproof of the  long standing “Telescope Conjecture ”, and explain how these can be used to obtain new results about the asymptotic behavior of the stable homotopy groups of spheres.