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 Kahler-Einstein edge metrics, Santalo-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.
Robert J. Berman, Chalmers University of Technology and University of Gothenburg, Sweden. Yanir A. Rubinstein, University of Maryland, College Park, MD.
Articles Chenzi Jin, The Cheltsov-Rubinstein problem for strongly asymptotically log del Pezzo surfaces Vlassis Mastrantonis, A Santalo inequality for the $L^p$-polar body Frederic Campana, Junyan Cao and Mihai Paun, Subharmonicity of direct images and applications Tsz On Mario Chan and Young-Jun Choi, An application of adjoint ideal sheaves to injectivity and extension theorems Tai Terje Huu Nguyen and Xu Wang, A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem Dror Varolin, Curvature of smooth proper direct images by way of a holomorphic Gauss formula Sebastien Boucksom and Mattias Jonsson, Measures of finite energy in pluripotential theory: A synthetic approach Jakob Hultgren, Duality of Hessian manifolds and optimal transport Laszlo Lempert, To the geometry of spaces of plurisubharmonic functions on a Kahler manifold Julius Ross and David Witt Nystrom, Harmonic interpolation and a Brunn-Minkowski theorem for random determinants Mingchen Xia, Operations on transcendental non-Archimedean metrics
