On smooth categorical compactifications and non-commutative Hodge theory

In this talk we disprove two conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin). In particular, we show that there exists a minimal 8-dimensional $A_{\infty}$-algebra, for which the supertrace of $m_3$ on the second argument is non-zero.
As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification (i.e. it cannot be represented as a quotient of a smooth and proper DG category). This gives a negative answer to a question of Toen. We also obtain an example of a proper DG category, which does not have a categorical resolution of singularities (i.e. it cannot be embedded into a smooth and proper DG category).