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Monday, September 17, 2018

**Abstract:** Pre-Calabi-Yau structures are certain structures on associative algebras introduced by Kontsevich and Vlassopoulos. This incorporates as special cases many other algebraic structures of diverse origins. Elementary examples include double Poisson algebras introduced by Van den Bergh, as well as infinitesimal bialgebras studied by Aguiar. Other examples also arise from symplectic topology as well as from string topology, whose relation with topological conformal field theory can be formulated in terms of pre-Calabi-Yau structures. In this talk, we will define pre-Calabi-Yau structures, and study it in the context of noncommutative algebraic geometry. In particular, we show that Calabi-Yau structures, introduced by Ginzburg and Kontsevich-Vlassopoulos, can be viewed as noncommutative analogue of symplectic structures. Pushing this analogy, one can show that pre-Calabi-Yau structures are noncommutative analogue of Poisson structures. As a result, we indicate how a pre-Calabi-Yau structure on an algebra induces a (shifted) Poisson structure on the moduli space of representations of that algebra.