Leo Herr Mangled Modules: Sheaves, Exactness, and perversion

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Given a multiplication structure on a set, if we enlarge the set a little, how many new multiplications can we make that extend the old one? Metaphor: If we fatten a ring slightly, can we also fatten a given module? The principle of this talk is examples; the content is my own research with Jonathan Wise. I will define two words: "sheaf" and "exact," and explain what it is, exactly, that I do here. It should rely only on groups, rings, modules, and topological spaces, but more will be helpful.