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In geometric Langlands, you start with a curve X, and try to put into correspondence two very different-looking entities: Local systems of rank n on X on the one hand, and Hecke eigensheaves on Bun_n(X) on the other. In the recent work of Fargues-Scholze, there is a beautiful conjectural extension of this story to the p-adic setting. In that setting, the role of X is played by the Fargues-Fontaine curve, and the local systems are Galois representations.

The conjecture of Fargues-Scholze has impressive explanatory power when it comes to the local Langlands program. For instance, it implies all cases of the Kottwitz conjecture on the cohomology of local shtuka spaces. We only know certain cases of that conjecture, notably for the Lubin-Tate tower (Harris-Taylor). We will discuss work with Hansen and Kaletha detailing partial results on the Kottwitz conjecture.

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