|Location:||MSRI: Simons Auditorium|
In the chromatic approach to stable homotopy theory, one computes various localizations of a finite spectrum then attempts to reassemble them to get the original spectrum. The Telescope Conjecture says that the localizations are computing when you hope they're computing, and the Splitting Conjecture gives a recipe of the reassembly. Both are
open and, indeed, almost untested. I'll explain both, give the cases for and against, and try to explain some approaches for tackling them.