Abstract: One of the oldest problems in stable homotopy theory is simple: just compute the stable homotopy groups of spheres. This turns out to be difficult, and a complete answer may never be known, but the computations continue. A recent technique being applied to great success can be summarized as: don't just compute the stable homotopy groups of the sphere spectrum, also compute the stable homotopy groups of other sphere spectra. These other sphere spectra include, for example, those arising in motivic and equivariant contexts, and can be thought of as deformations of the classical sphere spectrum. The homotopy theories they live in can then be thought of as deformations of classical stable homotopy theory. There are a few methods for building these deformations; the goal of this sequence of two talks is to describe the concrete approach to this deformation story using filtered objects. Please email vb8 at illinois dot edu for the zoom details.