This volume presents a collection of research articles arising from the conference on “Convex and Complex: Perspectives on Positivity in Geometry,” held in Cetraro, Italy, from October 31–November 4, 2022. The conference celebrated the 70th birthday of Bo Berndtsson and the vitality of current research across complex and convex geometry, as well as interactions between the two areas, all united by the overarching concept of positivity.

Positivity plays a central role in complex and convex geometry. It arises from a range of complementary perspectives, as illustrated by the breadth of the papers appearing in this volume, including existence Kähler–Einstein edge metrics, Santaló-type inequalities, curvature of direct images of bundles, extension theorems for holomorphic functions, optimal transport and Hessian manifolds, interpolation and Brunn–Minkowski theory, and non-Archimedean geometry. The format of the workshop was innovative compared to standard conferences in mathematics, with focused 30-minute talks, aimed at stimulating lively discussions and a “flipped classroom” where the audience becomes more engaged and the speaker is not expected to transmit more information than listeners can possibly absorb. Lengthy breaks between talks and a relatively small number of talks allowed for useful time blocks for collaboration.

This volume reflects the spirit of the conference, showcasing the vitality of current research in these areas as well as the profound impact of Bo Berndtsson's contributions to the field.