We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery and Tao. The proof uses the polynomial method introduced by Dvir.

We introduce the$generic central character$of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types ${E_{n}^{(1)}}$ , n=6, 7, 8) with$arbitrary positive parameters$and we prove an explicit product formula for their formal degrees (in all cases).

Consider the indicator function$f$of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of$f$is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension ${\frac{31}{36}}$ almost surely, and the corresponding dimension in the half-plane is ${\frac{5}{9}}$ . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.

We show that smooth maps are$C$^{1}-dense among$C$^{1}volume-preserving maps.

In the article “Distinguished varieties”, Lemma 4.6 was incorrect. We prove a different lemma that can then be used to prove the original Theorem 4.1.