While investigating fundamental questions in transcendental number theory, we have realized the need for a better understanding of basic positivity questions in Arakelov geometry. This has led us to a long undertaking of new fundations for parts of the subject, that would allow for both new results and concepts, and a significant simplification of and generalization of known results around arithmetic ampleness, which gives them much more flexibility in applications. Specific works related to this subject comprise:
-new equidistribution results on Shimura varieties, with applications to K3 surfaces (Tayou, Math Research Letters, Crelle, Forum of Math. Pi)
-an arithmetic Bertini irreducibility theorem (Charles, ASENS) and versions of an arithmetic Bertini theorem (Wang, JEP)
-new proofs and generalizations of arithmetic Hilbert-Samuel (Ni, 2 preprints)
-a monograph introducing theta-invariants for infinite-dimensional vector bundles (Bost, Progress in Mathematics (Birkhaüser))
-the development of geometry of numbers in infinite rank, with applications to a new class of geometric objects called A-schemes, giving a way of dealing with the cohomology of coherent sheaves in Arakelov geometry, and replacing classical methods of L2 analysis by "softer" functional analysis of nuclear spaces, with applications to arithmetic ampleness, affine objects in Arakelov geometry, Fekete-like theorems in higher dimension, and approximation results for holomorphic functions (4 manuscripts in finalization, totalling roughly 1000 pages)
-the introduction of the notion of formal-analytic arithmetic surfaces, with applications to fundamental groups and algebraization theorems (preprint to be published as a monography).
Regarding other aspects of the projects, both Bost and Charles have given Bourbaki seminars on topics that should play an important role (stability conditions and geometry of numbers). With Cadoret (Algebraic Geometry) and Pirutka (preprint), Charles has investigated uniform boundedness phenomena over number fields. Charles also investigated topics in complex geometry, regarding rational curves on hyperkahler varieties (with Mongardi and Pacienza, Compositio Math.) and Hodge structures on complex tori (Math. Z.).