Activities This Week

In this second talk we prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital A_\inftycategories with respect to the corresponding classes of quasi-equivalences are all equivalent. As an application, we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital A_\inftyfunctors between them. This is a joint work with Prof. A. Canonaco and Prof. P. Stellari arXiv:1811.07830.

A polytope is called simplicial if all its proper faces are simplices. The celebrated g-theorem gives a complete characterization of the possible face numbers (a.k.a. f-vector) of simplicial polytopes, conjectured by McMullen ’70 and proved by Billera-Lee (sufficiency) and by Stanley (necessity) ’80. The latter uses deep relations with commutative algebra and algebraic geometry. Moving to general polytopes, a finer information than the f-vector is given by the flag-f-vector, counting chains of faces according to their dimensions. Here much less is known, or even conjectured.

I will discuss what works and what breaks, at least conjecturally, when passing from simplicial to general polytopes, or subfamilies of interest.