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A thematic trimester in arithmetic will take place in ENS de Lyon and Université Lyon 1 from April 23, 2018 to June 29, 2018. This trimester is divided in two parts with different focus, with a conference taking place from May 22 to May 25.

The first part of the trimester focuses on the theory of algebraic groups, and particularly the Grothendieck-Serre conjecture on locally trivial homogeneous principal spaces. The conjecture was proved by Fedorov and Panin, following work of Colliot-Thélène, Nisnevich, Ojanguren, Ragunathan, Stavrowa, Vavilov...In the arithmetic case, Fedorov proved recently a significant special case of the conjecture. The study of this problem uses, among others, approximation techniques in algebraic groups, patching techniques, and affine grassmanians.

The second part of trimester focuses mainly on the question of the geometrization of the local Langlands correspondence. This problem was born from three major advances in arithmetic geometry: the introduction of the theory of perfectoid spaces by Scholze, the work of V. Lafforgue on the Langlands correspondence for function fields, and the introduction by Fargues and Fontaine of the fundamental curve of p-adic Hodge theory. We will take stock of the latest advances.