## Anton Leykin

This talk will start with an introduction to the homotopy continuation methods used in numerical algebraic geometry. The algorithms that are currently used are based on numerical heuristics: in general their results are not certified. Jointly with Carlos Beltran, using recent developments in complexity analysis rooted in the alpha theory of Smale, we have implemented a homotopy tracking algorithm that provides the status of a mathematical proof'' to its approximate numerical output.

## Claudiu Raicu

Any map of schemes $X\to Y$ defines an equivalence relation $R=X\times_Y X\to X\times X$, the relation of being in the same fiber". Koll\'ar asked whether all finite equivalence relations have this form (are effective). The answer to this question is in general negative, but is affirmative in the case of affine toric equivalence relations on affine toric varieties. I will explain the relationship between this result and the vanishing of the first cohomology group in the Amitsur complex associated to a toric map of toric algebras, and present a method for generating examples of noneffective equivalence relations in the nontoric case.