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Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful 'philosophical' principle that some mathematicians of the XIXth century seem to be well aware of. In general terms we would say that analytically unsound manipulations provide correct answers when they have an underlying transalgebraic background." Do you know more?

Edit: This text tells a few words more (e.g. "This philosophy can be linked to Kronecker’s ”Judgendtraum” and Hilbert’s twelfth problem, which seems to have remained largely misunderstood.") and refers to a manuscript "Transalgebraic Number Theory". Has someone a copy?

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here is the abstract of that talk: "Galois' visions contained in his brouillons and manuscripts goes far beyond the classical algebraic and differential Galois Theory. The goal pursued by Galois was to classify not only algebraic numbers and functions, but also transcendental ones. This philosophy goes far beyond our actual knowledge. We will present some elements of transalgebraic function field theory, as well as transalgebraic applications to the theory of Riemann zeta function." – M T Oct 23 '11 at 11:22
Thanks, Mattew! That makes one even more curious. I have not yet read the article linked to above and made my mind about it, but it looks as if there should be connections to periods, motives, F_un, etc. (?) – Thomas Riepe Oct 23 '11 at 11:54
Thomas, Could you check the links in your question? Some of them do not work, others apparently contain nothing related to your question. – Alexandre Eremenko Jun 30 '13 at 16:17
I found this texts: , (my computer does not produce a readable thing out of this:) , (book on "log-Riemann surfaces"): . – Thomas Riepe Jun 30 '13 at 20:16
MathSciNet says, No publications results for 'Anywhere=(transalgebraic)' – Gerry Myerson Feb 16 '15 at 22:22

For the most part "transalgebraic theory" seems an umbrella term for anything that relates (classic, differential, motivic, categorical) Galois theory with periods, trascendence results, special values, regularization...

Here is a compilation of papers which might be of interest.

Yves André

Michel Waldschmidt

Alexandru Buium

Alain Connes

Ricardo Pérez-Marco

  • The "Transalgebraic Number Theory" manuscript still doesn't seem to be avaible online. The author is clearly active (papers on arXiv), so perhaps someone can ask him directly.

  • The (only?) relevant online paper was alredy mentioned in the question, Notes on the Riemann Hypothesis.

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There is also this on the Arxiv:, Log-Riemann Surfaces by Kingshook Biswas and Ricardo Perez-Marco. – Gerry Myerson May 19 at 1:26

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