Day 2 (April 7) @ 11:00–12:30

Diletta Martinelli (University of Amsterdam)

On the birational classification of algebraic varieties

The classification of algebraic varieties is one of the central questions and guiding problems in algebraic geometry. For smooth complex projective curves we have a very satisfactory classification in terms of the genus that leads to three main types of curves: the projective line, elliptic curves, and curves of general type. I will explain how, thanks to the so-called Minimal Model Program, it is possible to recover this trichotomy for higher dimensional varieties and obtain three main pure types: Fano varieties, Calabi-Yau and canonically polarised varieties. I will conclude explaining some current research questions inspired by the geometry of Fano varieties.

Annegret Burtscher (Radboud Universiteit)

On the metric structure and convergence of spacetimes

Riemannian manifolds naturally carry the structure of metric spaces, and standard notions of metric convergence interact with the Riemannian structure and curvature bounds. No such profound theory is yet available for Lorentzian manifolds because the Lorentzian distance does not give rise to a metric structure. For spacetimes with suitable time functions, however, the recently introduced null distance provides an alternative metric that naturally interacts with the causal structure and yields an integral current space. In this talk we compare the metric space structures of Riemannian and Lorentzian manifolds and show how the convergence of a sequence of warped product spacetimes can be studied using the null distance.

Alessandra Cipriani (Delft University of Technology)

Random interfaces and Liouville quantum gravity

In this talk, we will introduce random interfaces in the context of conformal field theories. We will focus on the Gaussian free field (GFF), which appears as scaling limit of different statistical mechanics models and has been central in defining two-dimensional Liouville quantum gravity. We will review the most important results on the Liouville measure constructed as an exponential of the GFF, the study of thick points of such measure, and finally extensions of this construction in four (or rather, even) dimensions via another random interface, the membrane model.