European Congress of Mathematics
Amsterdam, 14–18 July, 2008
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Non-archimedean integrals are ubiquitous in various parts of mathematics. Motivic integration allows to understand them geometrically and to get strong uniformity statements. In these notes, intended for a general audience, we start by giving various examples of situations where one can get new geometric results by using p-adic or motivic integrals. We then present some more recent results in this area, in particular a Transfer Principle allowing to transfer identities involving functions deﬁned by integrals from one class of local ﬁelds to another. Orbital integrals occurring in the Fundamental Lemma of Langlands Theory form a natural family of functions falling within the range of application of this Transfer Principle.
Keywords: Motivic integration, arcs, p-adic integration, birational geometry, Milnor ﬁber