math.HO — History and Overview

The 57 Grothendieckian Problems: Open Problems for Mathematics and Machine Reasoning from the Grothendieckian Programme

Contributed by Vincent Liu

We state fifty-seven open problems drawn from the seven works of the Grothendieckian Programme (34 volumes, 26,021 pages, released openly in October 2026 as first drafts that have not been peer-reviewed). Thirty-five come from the two works on the architecture of a reasoning system and twenty-two from the five works that each take a named conjecture or programme as research subject; no problem is one of those conjectures. Each problem is stated in the words of its source, located by work, chapter and page, given the status the source assigns it, and paired with a criterion that decides when it is solved, a note on existing work and the date of a literature check. The problems are of three kinds: twenty-five mathematical problems, nineteen machine problems with success criteria fixed in advance, and thirteen formalization problems for Lean 4 and Mathlib. Every quotation from the sources has been checked against the published PDFs, and each problem has been checked against the literature as of 28 October 2026; a machine-readable version accompanies the paper.

Author comments: 54 pages, 5 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734427 (all versions: 10.5281/zenodo.23734426). This copy was typeset with pdfLaTeX for Hexagon: its text is identical to version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX, and a few line and page breaks differ. Each problem has a public issue at https://github.com/AlienMuse/grothendieckian-problems, which also holds the machine-readable data, the literature-check records and the Lean statement of Problem 49. Corrections recorded since version 1.0, including literature missed by the check for Problem 18: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Grothendieckian Programme: An Overview (DOI 10.5281/zenodo.23734421) and The Rising-Sea Theses (DOI 10.5281/zenodo.23734423).