Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of p-adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier's interpolation theorem.

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Under an assumption on the existence of $p$ -adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\text{big}~\mathbb{T}$ ’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where $n=2$ and  $p$ splits completely in the number field, we relate our construction to the $p$ -adic local Langlands correspondence for  $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ .

Let $f$ f be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over $\mathbf{Q}$ Q .

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.