Calendar

Just as moments of a distribution of real numbers are a powerful tool that (if they are not too large) determine the distribution uniquely, we can consider certain averages as moments of a distribution of groups (or more general algebraic structures).  We discuss how these moments determine a distribution of groups uniquely, and how that can be applied to find distributions of fundamental groups of random 3-manifolds, fundamental groups of curves over finite fields, and to give conjectures for distributions of important groups arising in number theory such as class groups.

Zoom:   https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

We prove that the categories of coherent (or, equivalently, of quasi-coherent) sheaves over two noetherian non-commutative schemes X and Y are equivalent if and only if there centers C(X) and C(Y) are isomorphic and there is a local progenerator in the category of coherent sheaves over X whose sheaf of endomorphisms is anti-isomorphic to the inverse image of the structure sheaf of Y under the isomorphism X->Y. To prove it, we combine the classical Morita theorem with the Gabriel’s theory of locally noetherian categories.