Abstract.

Pre-Calabi-Yau structures are certain natural structures on associative
algebras introduced recently by Kontsevich and Vlassopoulos. As special cases they incorparate many familiar algebraic structures of diverse origin. Elementary examples include double Poisson algebras introduced by Van den Bergh, as well as infinitesimal bialgebras studied by Aguiar. Other (more geometric) examples arise from symplectic topology and string topology.

In this talk, we will define pre-Calabi-Yau structures, and study
them in the context of noncommutative algebraic geometry. In particular, we will show that, while classical Calabi-Yau structures (in the sense of Ginzburg) can be viewed as noncommutative analogue of symplectic
structures, the pre-Calabi-Yau structures are natural noncommutative
analogues of Poisson structures. As a main theorem, we prove that a
pre-Calabi-Yau structure on an algebra induces a (shifted) Poisson structure on the (derived) moduli spaces of representations of that algebra.