Periodic Reporting for period 1 - DiophGeo (Diophantine Geometry: towards the ultimate Bogomolov conjecture) Reporting period: 2021-04-01 to 2023-03-31 Summary of the context and overall objectives of the project Diophantine equations are equations in which solely addition, subtraction, multiplication, whole numbers, and a finite number of unknowns appear. Since antiquity, mathematicians have tried to determine the solutions of such equations in whole numbers or integers. The results of the project (proofs of the uniform Bogomolov and Mordell-Lang conjecture) have provided us with much better "geometric" tools for this quest, though many open problems remain in this long-running cultural endeavor of mankind spanning several centuries. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far (1) The main results of the project have been strong uniform results of Manin-Mumford, Bogomolov, and Mordell-Lang type. As yet, the results have been submitted and are available as preprints.(2) The extension to a relative Bogomolov conjecture has been explored, but only proven for fibered products of families of elliptic curves. The results are published in Journal für Reine und Angewandte Mathematik.(3) In addition, the Pink-Zilber conjecture on unlikely intersections for curves in semiabelian varieties has been proven and the proof published in a research article in Selecta Mathematica. As a continuation, new ideas for its proof beyond the case of curves have been obtained. (4) Finally, the Investigator taught 2 student seminars, 1 lecture, supervised a master student, and co-supervised a PhD student at KU, improving his track record in teaching. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The work on uniformity in diophantine geometry conducted during this project has led to long-expected, but previously unproven results that will shape future research in diophantine geometry and related areas (e.g. arithmetic dynamics). Though this will shape future research in the field, no wider societal implications are expected. a (happy) family of abelian varieties
