# Patching and $p$-adic local Langlands

 Joint IAS/Princeton University Number Theory Seminar Topic: Patching and $p$-adic local Langlands Speaker: Ana Caraiani Affiliation: Princeton University; Veblen Research Instructor, School of Mathematics Date: Thursday, December 5 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University

The $p$-adic local Langlands correspondence is well understood for $\mathrm{GL}_2(\mathbb Q_p)$, but appears much more complicated when considering $\mathrm{GL}_n(F)$, where either $n>2$ or $F$ is a finite extension of $\mathbb Q_p$. I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the p-adic local Langlands correspondence for $\mathrm{GL}_n(F)$ using global methods. The key ingredient is Taylor-Wiles-Kisin patching of completed cohomology. This allows us to prove many new cases of the Breuil-Schneider conjecture.