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IMS Distinguished Visitor Lecture Series
Pseudo-finite Fields, Motives and Integrals

François Loeser
Sorbonne University, France

Pseudo-finite fields were introduced by J. Ax as a first-order axiomatization of finite fields. Together with J. Denef, we assigned to a definable set over a pseudo-finite field a geometric object, called a virtual Chow motive, that encapsulates its number of points. This device can be used to construct a motivic measure for definable sets over nonarchimedean local fields for which a generalization of the classical AxKochen Theorem holds. More generally, together with R. Cluckers we proved a general transfer principle
allowing to transfer identities between functions defined by integrals over nonarchimedean local fields of characteristic zero to local fields of positive characteristic and vice versa. As we proved together with R. Cluckers and T. Hales, this principle can be for instance applied to the Fundamental Lemma of Langlands-Shelstad, an identity between orbital integrals whose proof by Ngô achieved considerable fame. More recently we have been able with A. Forey and D. Wyss to use some of these methods to obtain a motivic
enhancement of this identity.