Appears in collection : 2023 - T2 - WS2 - Higher structures in enumerative geometry

In derived algebraic geometry, vector bundles can be shifted. This corresponds to the shearing operation on graded objects. However, over the sphere spectrum, the notions of shifting vector bundles and of shearing gradings decouple, in parallel to the existence of two variants of the affine line. In this talk, we will discuss how the picture of shifting and shearing can be reconciled by replacing the integers with more exotic grading groups. We will see how this naturally leads to considering the moduli stack of oriented formal groups, and through it to chromatic homotopy theory and the motivic tau-deformation phenomenon.