[Math] Durov approach to Arakelov geometry and $\mathbb{F}_1$

Question

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry over the tropical semiring. The analytic notions in Arakelov geometry seem to appear naturally, as a consequence of rephrasing the construction of $\mathbb{Z}_p$ in terms of maximal compact subgroups of $GL(n, \mathbb{Q}_p)$, and carrying this construction to $GL(n, \mathbb{R})$. A proposed construction of the field with one element $\mathbb{F}_1$ and its finite extensions also appear in the thesis.

Given these premises, I'd want to understand more of Durov's work. But Durov has left mathematics to found Telegram, and I am not sure about the status of his work. Since my background on arithmetic geometry is limited, and I am not currently in the university, I have troubles evaluating the impact of this approach, and I would like to know more before delving into a 568 pages thesis.

Has this theory been developed after he left mathematics? Did anyone find applications outside the theory itself? What is the point of view of people working in "classical" Arakelov geometry?

EDIT I had incorrectly assumed that after founding Telegram, Durov had left mathematics, but as @FedorPetrov points out, he is still active. Yet the question still stands: what is the status of this approach to Arakelov geometry? Is he (or other people) still developing it? Were there any results unrelated to generalized rings proved using this theory? I do not have access to his recent papers, but judging from the first page, his focus seems to have shifted somehow

Answer 1

Let me just give a quick list of references and some brief comments:

• for a survey of the various approaches to $\mathbb{F}_1$ see the excellent paper "$\mathbb{F}_1$ for everyone" by Lorscheid (doi:10.1365/s13291-018-0177-x, arxiv:1801.05337);
• the $K$-theory of generalized rings has been investigated by Scholbach (doi:10.1007/s40062-014-0085-4, arxiv:1202.5203, PDF);
• some foundational work on generalized rings has been done by Hablicsek and Juhász (doi:10.1080/00927872.2017.1344691, arxiv:1701.02178).

There is a lot of work to do with generalized rings, especially concerning $K$-theoretic matters... All the usual approaches (via exact sequences and the like) seem to have failed to capture the information which is contained in these rings...

This being said, I am absolutely not working in this field, and would like to know a lot more myself! Thanks for reading!