**Local to global in modular representation theory and homotopy**

Many questions in mathematics evolve around passing from
« local » information to « global » information. In modular representation
theory
long-standing conjectures predict how representations of G relate to
representations of « local » subgroups, though the exact nature of the
proposed bijection is often mysterious. One of the strengths of
homotopy theory is that it can allow for more creative sorts of
induction, or gluing, taking into account also higher order structure.
My talk will describe one success of this viewpoint, in the
classification of so-called endotrivial modules, which can be though
of as « almost-1-dimensional » modules. I will tell this story from the
beginning, starting with work of Dade in the 70s, and leading into the
present…