The team has worked on all the 4 birational objective mentioned above, and also on many of the additional and application objectives. The team could solve 3 of the 4 birational objectives, most application and additional objectives, and even newer questions that were not in the original proposal. However, we were not able to get hold of the birational objective "locally stable reduction". This is a famous unsolved problem that many have tried to solve, but so far it has been eluding everyone, including the team.
The "MMP for arithmetic 3-folds" objective was resolved in the article “Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic" by Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi (team member), Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek. This article appeared in the Publications Mathématiques de l'IHÉS.
The objective "Gluing theory for surfaces" was showed in the articles by Quentin Posva (team member) titled "Gluing theory for slc surfaces and threefolds in positive characteristic", "Gluing for stable families of surfaces in mixed characteristic" and "Abundance for slc surfaces over arbitrary fields".
The objective "Local Kawamata Viehweg vanishing" was showed in the article "On the properness of the moduli space of stable surfaces over ℤ[1/30]" by Emelie Arvidsson, Fabio Bernasconi and Zsolt Patakfalvi (team member).
Work in the direction of the application/additional objectives were considering the following topics.
1.) Wall crossing property of the moduli space of stable varieties in the article "Wall crossing for moduli of stable log varieties" by Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro and Zsolt Patakfalvi (team member). This appeared in the Annals of Mathematics.
2.) Alternative construction of the moduli space of stable varieties in the article
"Moduli of Q-Gorenstein pairs and applications" by Stefano Filipazzi (team member) and Giovanni Inchiostro.
3.) Geometry of Calabi-Yau varieties by the team member Stefano Filipazzi with the articles "Baily--Borel compactifications of period images and the b-semiampleness conjecture" (with Benjamin Bakker, Mirko Mauri and Jacob Tsimerman) and "Boundedness of some fibered K-trivial varieties" (with Philip Engel, François Greer, Mirko Mauri and Roberto Svaldi).
4.) Positive characteristic geometry by the team members Jefferson Baudin, Zsolt Patakfalvi and Javier-Carvajal Rojas. Sample articles are "A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves" by Jefferson Baudin, and the article "The Demailly--Peternell--Schneider conjecture is true in positive characteristic" by Zsolt Patakfalvi and Ejiri Sho.
5.) Mixed characteristic geometry by the team member Zsolt Patakfalvi. For example the article "Test ideals in mixed characteristic: a unified theory up to perturbation" (with Bhargav Bhatt, Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek) gives a systematic treatment of the mixed characteristic test ideals (the mixed characteristic versions of the multiplier ideals).