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Adelic Geometry via Topos Theory (note change in time!)

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If you have a question about this talk, please contact mxn732.

This is a practice run for a talk I will give at Toposes Online at the end of the month. Official duration of the talk is 25 minutes, with 5 minutes at the end for questions—I’d like to use this as an opportunity to figure out how I should pace myself.

Abstract: In this talk, I will give a leisurely introduction to the theory of classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory. In more detail: recall that every Grothendieck topos classifies some geometric theory T and possesses a generic model U_T that is conservative, i.e. any property P expressible as a geometric sequent is satisfied by U_T iff  P is satisfied by all models of T. One can thus ask the following question: can we construct a classifying topos whose points (i.e. models) are the completions of Q? In particular, this would give us a framework for reasoning about properties that hold for the generic completion of Q (and by extension, properties that hold for all completions of Q as well)—much like how the adele ring A_Q of Q is a device that allows us to reason about all the completions of Q at once.

The first step towards constructing such a topos is to define the geometric theory of absolute values of Q and provide a point-free account of exponentiation—this has already been completed in [2]. The next step is to construct the classifying topos of places of Q, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of Spec(Z). This part is still work in progress, but some interesting observations (in particular, regarding quotienting and point-set reasoning) have already emerged which we would like to share with the community.

[1] Adams, JF, Fiedorowicz, Z. Localisation and Completion, arXiv:1012.5020.

[2] Ng, M, Vickers, S. Point-free Construction of Real Exponentiation, arXiv:2104.00162.

[3] Mazur, B., On the Passage from Local to Global in Number Theory, Bulletin of the AMS , Volume 29 No. 1, (1993), pp. 14,51

This talk is part of the Theoretical computer science seminar series.

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