## Symmetries and correspondences |
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Talks and discussions

Abstract. I will explain an approach to constructing geometries relative to a symmetric monoidal category. I will then introduce the category of normed sets as a possible analytic geometry over the field with one element. I will show that the Fargues-Fontaine curve from p-adic Hodge theory and the Connes-Bost system are naturally interpreted in this geometry.

14:30-16:00 **Boris Zilber**
*Algebraically closed fields of characteristic one*

Abstract. I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction leading to a a 'pseudo-analytic' structure which we identify as an algebraically closed field of characteristic 1. Then will formulate precise axioms that such a field must satisfy. The main theorem then states that under these axioms the structure has the desired algebraic and analytic properties. The axioms have a form of statements about existence of solutions to systems of equations in terms of a multi-dimensional valuation theory and the validity of these statements is an open problem to be discussed.