Tom Church

Abstract: Representation theory over Z is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover, it turns out that important questions in topology / number theory / representation theory / ... correspond to asking whether familiar algebraic properties hold for these "rings". I'll explain how these connections work; describe what we know and don't know; and give a wide sampling of concrete applications in different fields. No knowledge of representation theory will be required -- indeed, that's sort of the whole point!