Since the PI's arrival in Europe, the major new breakthrough in the project has come from the collaboration of the PI with his new colleagues in Milano.
The colleagues, Paolo Stellari at the University of Milan and Alberto Canonaco at the nearby University of Pavia, have spent much of the last decade and a half on the study of dg enhancements of triangulated categories, and of their uniqueness. And the big surprise, which the three of us discovered together, is that this can be combined with the seemingly-unrelated techniques that the PI developed, back in Australia, before the start of this ERC project. We have one article already submitted, a preprint on the archive, and another being completed.
In more detail: our (submitted) preprint of 2024, meaning a joint paper with Canonaco and Stellari, gives a thorough exploration of the intricate relationship between the various intrinsic subcategories of weakly approximable triangulated categories. This amounts to an extension of the new theory started in Australia, and so far makes no mention of enhancements.
A very long preprint (96 pages), which I wrote in 2025, studies when the completion of a triangulated category with respect to a good metric is involutive. The class of metrics for which this happens is called the excellent metrics.
These are the two preprints that already exist on the archive, and so far neither of them uses or even mentions enhancements. They were written to prepare the ground for the third paper, which does use enhancements and prove results about them.
And finally we come to the most surprising: by combining the results of these two recent preprints, with the 2022 article that I wrote with Alberto Canonic and Paolo Stellari
* "Uniqueness of enhancements for derived and geometric categories". Forum of Mathematics, Sigma 10 (2022) Paper No. e92, 65 pages.
one ends up not only much extending the results of the 2024 preprint above, but also proving new uniqueness-of-enhancements theorems. And the techniques are unexpected and novel.
And finally I should mention the efforts to spread the word and get others interested.
The first example is the progress in the PhD thesis of my student Kabeer Rahul Manali, who made giant steps forward in generalizing my old, Australian work. Kabeer's thesis uses a class of metrics more general than the one I worked with, and shows that many of the results I proved can be extended to this greater generality.
In the last year or two there has been a string of articles by other authors, in Europe but also in China and in Russia, who have independently become involved in this new theory - in part probably because of my efforts in advertising the work. There are two many preprints and authors to list them all here.
And undoubtedly the effort to promote this new theory will a giant boost very soon, when the next major article in the series will appear in Inventiones Mathematicae. The article was accepted on December 14, should appear in the coming months, and young people tend to pay attention to articles that appear in very prestigious journals like Inventiones.
