Recently in a joint work with J. Dobrowolski and N. Ramsey it is shown that in any NSOP1 theory with existence,
Kim-independence satisfies all the basic axioms over sets (except base monotonicity) that hold in simple theories with forking-independence. This is an extension of the earlier work by I. Kaplan and N. Ramsey that such hold over models in any NSOP1 theory. All simple theories; unbounded PAC fields; vector spaces over ACF with bilinear maps; the model companion of the empty theory in any language are typical NSOP1 examples.

   An important issue now is to know the existence of canonical bases. In stable and simple theories well-behaving notion of canonical bases for types over models exists, which is used in almost all the advanced studies. But there are a couple of crucial obstacles in finding canonical bases in NSOP1 theories. In this talk I will report a partial success/limit of the project. Namely, a type of a certain Morley sequence over a model has the weak canonical base. In my talk I will try to explain all the related notions.

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.