In planar algebras, we show how to project certain simple quadratic tangles onto the linear space spanned by linear and constant tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

We generalize the decomposition theorem for perverse sheaves to Artin stacks with affine stabilizers over finite fields.

We strengthen the local-global compatibility of Langlands correspondences for ${GL}_n$ in the case when $n$ is even and $l\ne p$. Let $L$ be a CM field, and let $\Pi$ be a cuspidal automorphic representation of ${GL}_n\left({\mathbb{A}}_L\right)$ which is conjugate self-dual. Assume that ${\Pi }_{\infty }$ is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_l(\Pi )$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of $R_l(\Pi )$ to the decomposition group at $v$ corresponds to the image of ${\Pi }_v$ via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that $\Pi$ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator $N$ on the complex of nearby cycles on a scheme which is locally étale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for $\Pi$ as above.

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.