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Coalgebras, deformation theory, and Koszul duality

Coalgebras are the opposite of algebras: instead of a multiplication they have a comultiplication. While this may seem like a superficial difference, coalgebras and algebras are actually quite different sorts of object. However, up to a suitable notion of homotopy, one can view coalgebras as algebras, and vice versa: this is the subject of Koszul duality. I'll give an introduction to coalgebras (with plenty of examples) and skim a little bit of the theory of Koszul duality. Time permitting, I'll explain what any of this has to do with infinitesimal deformations of objects from (derived) algebraic geometry. This will be loosely based on a paper of mine from 2022 together with work currently in progress joint with Andrey Lazarev.