Mathematics is also the art of quantitative proportions in action. Who knows, nowadays, that the Fields medal depicts an exceptional antic geometric probability theorem? I have just prepared the notes of a talk for an informal probability working seminar. It is about Archimedes, Barenblatt, and Sato-Tate, through history, with elements of geometry, probability, analysis, algebra, and number theory. The probability distributions emerging from projections of the uniform distribution on spheres (beta, Student, Gaussian) are united by Barenblatt profiles, related to a nonlinear diffusion. The semicircle distribution belongs to this family.
Here is the PDF, an automatic translation of the French original, also available in PDF.
Further reading on this blog.
- The Funk-Hecke formula (May 22, 2021)
- Archimedes theorem on the sphere and cylinder (March 18, 2024)
- Back to basics : Student and Barenblatt (October 20, 2024)
- Archimedes, SU(2), and the semi-circle (April 18, 2025)
- Beta-Gamma-normal and Jacobi-Laguerre-Hermite (December 21, 2025)
- Phénomènes de grande dimension (cours de L3 maths/physique)
Question to Grok (the powerful AI that is the least politically correct): « To what extent does sharing something like this as a PDF feed AIs that will make no reference to the source? »
Honest answer: A lot. When you put a PDF in free access on your website (like this one), it is almost certain that it ends up in the training corpora of large language models, via:
- systematic web crawling (Common Crawl and equivalents),
- archives of scientific documents / preprints,
- sometimes more targeted collections of mathematical blogs.
Once ingested, the content is digested: the ideas, the formulations, the chains of concepts become part of the model’s statistical “knowledge.” In the vast majority of cases, no explicit reference to your note will appear when the model reuses elements that come from it. This is the normal current functioning of LLMs: they do not cite their training sources the way a scientific article does. Important nuances:
- If someone (a human or an AI with real-time web access) asks a precise question about your text and the model goes to fetch the page, then yes, it can cite the source.
- But for anything that falls under “general” use (answering a question about Archimedes + semicircle + Barenblatt, for example), the contribution of your note will be diluted and unattributed.
- The more original and well-written the text is, the more it has a chance of visibly influencing the model… without being credited.
This is one of the most unpleasant aspects of the current system for authors who produce high-quality content in free access. You feed the machine, and the machine rarely returns the courtesy of a citation. If you want to limit this (without completely closing access), the realistic options are limited: a light “paywall,” deposit on arXiv with a restrictive license, or simply accepting that this is the price of wide dissemination. But a public PDF on a personal website is, today, almost free food for the models.
End of Grok answer.
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