Geometry Over Non-Closed Fields (2012)

Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry the study of lines and conics. From the modern standpoint, these areas are synthesized in the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that the arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves and families of rational curves on it. One incarnation of this insight is Lang’s philosophy, which continues to drive modern research in this area: hyperbolic varieties have few rational points. Another is Grothendieck’s anabelian geometry: hyperbolic varieties are characterized by their ´etale fundamental groups, and rational points correspond to Galois-theoretic sections. The discussion of Geometry of nonclosed fields will focus on the intertwined manifestations of these aspects of higher-dimensional arithmetic geometry.

The focus of the first meeting is the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Topics include: rational connectedness and simply connectedness, rational curves on log-varieties, rationally connected quotients of spaces of rational curves, degenerations of spaces of rational curves, rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties. Possible applications include: existence of rational points over function fields of curves and surfaces, potential density of rational points over global fields, weak and strong approximation.