Karol Palka

Title: Recent progress in the theory of almost minimal models

Abstract: 
Even for smooth log surfaces (X,D) minimal models can be singular, which makes many problems 
concerning them hard. If D is reduced there exists an "almost minimal model" which is log smooth, 
dominates the minimal one and can be directly constructed from (X,D) without introducing singularities. 
In 2014 we generalized this notion to boundaries with rational coefficients and later used it to prove 
the Coolidge-Nagata conjecture. We will report on recent progress (joint work with M. Koras and T. Pełka) 
in the problem of understanding planar curves homeomorphic to a line. We will also discuss important 
relations to log del Pezzo surfaces and the problem of bounding singularities of general planar curves 
with given homology.