T-invariant is publishing its latest regular report, Chronicle of the Persecution of Scientists No. 36, dated August 31, 2026. Azat Miftakhov has published a paper in the French quarterly La Revue de la filière mathématiques (June–August 2026). The paper was written at the suggestion of French mathematician Bernard Randé, a member of the journal’s editorial board and of the international association Solidarité FreeAzat. Miftakhov completed the work while held in prison in Dimitrovgrad.
Azat Miftakhov’s paper is titled “La conjecture de Chui dans un cadre hilbertien” (“Chui’s Conjecture in a Hilbert Space Setting”). Unfortunately, we have not been able to read it. The paper was not posted on arXiv.org, and the journal itself is available only by subscription. However, The Insider published the first page of Miftakhov’s manuscript, which gives a sense of both the subject of the paper and his approach to the problem.
The problem Miftakhov tackled concerns a variant of Chui’s conjecture. The conjecture came about in 1971, more or less by accident. In a short note, the well-known American mathematician Charles Chui asked how to place N unit point charges on the unit circle so that the average strength of the field they generate inside the disk is as small as possible. More precisely, he asked whether this quantity is bounded below by an absolute constant — one that holds for any number of charges in any arrangement. The answer came a year later: in 1972, Donald Newman published an elegant solution — yes, such a constant exists, and π/18 will do.
The conjecture itself is a remark Chui dropped in that same note. It is plausible, he wrote, that the minimum is attained by the most natural arrangement: the charges placed at the vertices of a regular N-gon. Chui offered no proof and moved on to entirely different things. The conjecture remains unproven to this day.
Even the case of two charges turned out to be hard. The proof was found in 2023 by Fedor Nazarov, a professor at Kent State University in Ohio. He posted his answer on MathOverflow, a forum where mathematicians put questions to one another, and where Nazarov has been answering his colleagues’ questions for many years under the username fedja.
Mathematicians have tried reformulating Chui’s conjecture for Hilbert spaces (that is, spaces equipped with an inner product). In this setting, there has been significant progress. In 2020, Evgeny Abakumov (Gustave Eiffel University, Marne-la-Vallée), Alexander Borichev (Aix-Marseille University), and Konstantin Fedorovskiy (Bauman Moscow State Technical University and St. Petersburg State University) posted an important paper, “Chui’s conjecture in Bergman spaces.” They proved Chui’s conjecture in weighted Bergman spaces for a broad class of weights: the minimum is attained if and only if the charges are evenly spaced. But some cases remained open, and the authors listed them explicitly at the end of the paper.
It was precisely this set of problems that Randé proposed to Miftakhov. The first page of his manuscript opens with the words “Let us assume” and contains the statement of the theorem and the beginning of the proof. He looks at the case where the charges lie strictly inside the disk rather than on the circle itself. And what he proves is not an inequality but an exact equality: the energy of any configuration equals the energy of the regular-polygon configuration plus an explicit nonnegative term that vanishes only when the charges form a regular polygon. If Miftakhov has indeed proved this, then Chui’s claim (for Bergman spaces) follows as a corollary. The original Chui conjecture, however, remains open.
On March 5, 2026, three mathematicians from the St. Petersburg Department of the Steklov Mathematical Institute — Evgueni Doubtsov, Anton Tselishchev, and Ioann Vasilyev — posted the preprint “Weighted Chui’s conjecture.” Their work follows Newman’s line of attack: they generalize his bound to arbitrary positive charges and to spaces of any dimension. At the end of the paper, the authors pose several open questions, the first of which is what happens when the charges lie not on the circle but inside the disk.
On March 6, 2026, Bernard Randé received Miftakhov’s manuscript, in which the charges likewise lie inside the disk. These are different problems, but the setup is the same — and so is the timing.
The Solidarité FreeAzat website writes: “In October 2025, Solidarité FreeAzat, together with the Navalny France team and the organization La Libre Pensée, held an evening in Paris devoted to writing letters in support of Russian political prisoners. Among the participants was Bernard Randé, a mathematician, a former teacher in the preparatory classes at the Lycée Louis-le-Grand, and a member of our association. Instead of an ordinary letter of support, he decided to give Azat what he misses most — a real mathematical problem.”
Getting mathematical formulas into facilities run by the FSIN [Russia’s Federal Penitentiary Service — T-invariant] is a challenge in its own right: such letters often never reach their recipients. A long wait followed, and on March 6, 2026, Bernard Randé received a reply from Azat via Solidarité FreeAzat. The letter opened with the words: “And so your problem, which brightened my days and cost me many hours of sleep, became a breath of fresh air for me. Thank you so much for that!” Attached to the letter were six handwritten pages containing the solution to the problem Randé had posed.
The Solidarité FreeAzat website writes: “Azat worked under exceptionally harsh conditions. Over four months in detention — with no scientific library, no internet access, no chance to discuss mathematics with colleagues, and only a few hours a day with pen and paper — he managed to solve a problem that was open at the time.” Here we have to take their word for it, since we have not yet seen Miftakhov’s paper. Still, the fact that the paper was published is a strong signal in itself.
Averil Aussedat, a member of Solidarité FreeAzat and a postdoctoral researcher in mathematics at the University of Pisa, proposed that her university officially invite Azat to present his result. On July 23, 2026, Azat Miftakhov’s wife, Elena Gorban, received an official letter from the University of Pisa inviting Azat to speak at a research seminar to be held in April 2027. If Miftakhov is not allowed to present his work in Pisa in April 2027, one can only hope that on September 3, 2027, when his latest sentence ends, he will be released and able to engage with colleagues and work under normal conditions.
Chui’s problem has been around for 55 years. Today it is being worked on by Abakumov in Paris, Borichev in Marseille, Fedorovskiy in Moscow, Doubtsov and his colleagues in St. Petersburg, and Nazarov in Ohio. Now one more participant is joining the conversation — from a prison in Dimitrovgrad, from the Polar Owl penal colony [a special-regime prison above the Arctic Circle — T-invariant], in a letter written on FSIN letterhead. Not as an object of pity or a media story, but as a colleague — with a new theorem.