Let $X$ be a smooth subvariety of a complex torus. For general data vectors the MLE problem on $X$ has exactly $|\chi(X)|$ solutions. We investigate what happens to these solutions when the data vector approaches a non-general value. Assuming that $X$ is schön we relate this behaviour to special rays in the tropical variety of $X$. We also explain how the non-general data vectors are related a Bernstein-Sato ideal associated to X. Based on joint work with Anna-Laura Sattelberger.

Derived stacks play the same role in the theory of moduli that projective resolutions play in the study of modules. In the work of Toen and Vezzosi, Lurie, and Pridham, derived stacks are realized as fibrant objects in a combinatorial model category. We prefer to think of derived stacks as the objects of a category of fibrant objects, the right fibrations of simplicial derived schemes with a fixed base $B_*$.

An important model for $\infty$-categories is Rezk's theory of complete Segal spaces. Their definition actually makes sense for simplicial objects in any category of fibrant objects. In joint work with Kai Behrend, we show that the complete Segal spaces in a category of fibrant objects are the objects of a category of fibrant objects if the category of fibrant objects satisfying an additional axiom:

If $f \colon X \to Y$ is a trivial fibration and $g \colon Y \to Z$ is a morphism such that $gf$ is a fibration, then $g$ is a fibration.

Many (if not all) categories of fibrant objects satisfy this axiom: for example, it holds for Kan complexes.

There is a functor from quasicategories to complete Segal spaces, which shows that these two theories are closely related. Boavida de Brito and Rasekh extend this to a functor from cartesian fibrations to complete Segal spaces in the category of right fibrations. Thus, the above result may be interpreted as a concrete realization of cartesian fibrations in derived geometry.

We describe a new model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We discuss the proof that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor, and a new theory of cones in cubical sets which is used in this proof. We also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts. This talk is based on joint work with Chris Kapulkin, Zachery Lindsey, and Christian Sattler, arXiv:2005.04853.