Unsolicited Nonsense

I wanted to quickly mention an “almost” LLM result. I wrote previously about a marvelous new approach of Yuchen Bi to Stoker’s conjecture using differential geometry, but that still critically used quite extensively non-polyhedral techniques. Yesterday, Arseniy Akopyan wrote to me that he chatted with LLM and found a polyhedral only proof.

Not essentially new, Arseniy and I then noticed that the LLM had “stolen” a result of Martin Winter, who in 2023 found the proof without noticing, and wrote it in a very nice paper expanding our knowledge of Wachspress coordinates and Izmestiev matrices. Congratulations, Martin!

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Recovering from some surgeries, I wanted to write a little something on Stoker’s conjecture: It is best viewed as an angular converse to Cauchy’s rigidity theorem. Cauchy’s proves more than the usual statement that congruent corresponding faces determine a convex polyhedron: for a fixed combinatorial type, the angles inside the faces already determine the dihedral angles. In 1968, James J. Stoker asked whether the converse holds: do the dihedral angles determine the face angles?

(Ps: For health reasons, I am taking a break and may not check or respond to communication)

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Misha Verbitsky, a distinguished mathematician and long-time public critic of the Russian state, has reportedly been detained at Yerevan airport on a Russian request.

Verbitsky is known not only for his mathematical work, but also for his uncompromising public writing: against war, against censorship, in favor of open culture and free expression. One need not agree with everything he has written to see the danger here. The Russian charges against him arise in the context of political speech and dissent. Extradition to Russia would therefore place him at grave risk.

Armenia should not hand him over. At minimum, Verbitsky must have immediate access to lawyers, independent observers, and a fair procedure in which the political nature of the Russian request is taken seriously.

This is urgent. Please circulate reliable information, contact academic and human-rights networks, and ask Armenian authorities not to extradite Misha Verbitsky to Russia.

For questions, please contact his daughter, Sima

see also change.org/free_misha_verbitsky

Update: he was freed and left Armenia

The one where some mathematicians meet on Lesbos, and observe that all F-pure standard graded Gorenstein rings have the hard Lefschetz property, but some Gorenstein domains do not.

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There is nothing new in this post. It is the sort of thing that most people would probably call folklore or obvious, but then again, sometimes it is cool to write it down. And who knows, perhaps someone finds it useful. Also… let’s face it, a lot of PL/combinatorial topology is a mess, and things that used to be well known get lost.

There are, on occasion, obvious concepts which lack a name. Perhaps because they have not proven useful yet. But then again, they are so very natural, they deserve to be discussed somewhere.

If X and Y are two topological spaces, perhaps manifolds, perhaps something else, then one remembers from topology what it means for them to be homeomorphic. We also know what it means to be PL homeomorphic¹, that is when the homeomorphism isn’t merely continuous, but piecewise linear. Still, it pays sometime to go back in time and look at the history of the subject. Incidentally, let me mention this marvelous book on the Hauptvermutung and its history I found recently by the late Yuli Rudyak².

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Next year we will have another combinatorics and geometry conference in Greece, organized with Enis Kaya, Evita Nestoridi, Stavros Papadakis, Vasiliki Petrotou and Christos Tatakis. Looking forward to seeing you there!

AI claims of achievement at the IMO aside, I found this really cool: One of the gold medalists (from China) won gold despite cerebral palsy

With Tosun, Mineev, and Tatakis. See you there 🙂

Firstly, for a while now, Harald Helfgott, Vasiliki Petrotou, Arina Voorhaar and myself have been organizing CAGe, a bimonthly (the bimonthly that happens roughly every 6 weeks) miniworkshop at IMJ in Paris. The idea was to have this instead of a seminar so that the visiting speakers could also have interesting talks to join, and so that there would be more of a theme each time. Next time is March 4, with Evita Nestoridi, Carsten Peterson and Jacinta Torres. Enjoy and join!

Secondly, after settling down at IMJ and getting a couch and carpet for my office, I will be teaching a course with Harald in the spring, on spectral graph theory. Find it here.

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