Date: 2 November 2018 Location: Hempfield High School, Room 213 Speaker: Irina Bobkova, IAS Princeton Title: Decomposition and duality in stable homotopy theory Abstract: Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology. This important problem has connections to many other areas of modern mathematics, including differential geometry, number theory and derived algebraic geometry. I will explain how chromatic homotopy theory allows to split this problem into easier pieces, which can be understood using the theory of formal group laws. I will talk about recent results, and work in progress, at the second chromatic level. I will describe a local version of Spanier--Whitehead duality and how the sphere spectrum at the second chromatic level can be decomposed in terms of simpler objects and their Spanier--Whitehead duals.