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Abstract:  There is a long history of networked dynamical systems that models the spread of opinions over social networks, with the graph Laplacian playing a lead role. One of the difficulties in modelling opinion dynamics is the presence of polarization: not everyone comes to consensus. This talk will describe work joint with Jakob Hansen [OSU] introducing a new model for opinion dynamics using sheaves of vector spaces over social networks. The graph Laplacian is enriched to a Hodge Laplacian, and the resulting dynamics on *discourse sheaves* can lead to some very interesting and perhaps more realistic outcomes.