We prove an upper bound on the optimal Hölder exponent for the chordal $\mathrm{SLE}$ path parameterized by capacity and thereby establish the optimal exponent as conjectured by Lind. We also give a new proof of the lower bound. Our proofs are based on sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.

Introduced in the late 1960s, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics that describes a system of interacting particles hopping left and right on a one-dimensional lattice of $n$ sites. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities $\alpha$ and $\gamma$, and they may exit and enter at the right with probabilities $\beta$ and $\delta$. In the bulk, the probability of hopping left is $q$ times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters $\gamma =\delta =0$. Using our first result and also results of Uchiyama, Sasamoto, and Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980s there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g., Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.

This paper establishes new estimates for linear Schrödinger equations in ${\mathbb{R}}^3$ with time-dependent potentials. Some of the results are new even in the time-independent case, and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's theorem.

We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schrödinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

Consider a normal complex analytic surface singularity. It is called Gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically Gorenstein if this line bundle is topologically trivial. The second notion depends only on the topological type of the singularity. Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic topological type. The question to know if any numerically Gorenstein topology was realizable by some Gorenstein singularity was left open. We prove that this is indeed the case. Our method is to plumb holomorphically meromorphic $2$-forms obtained by adequate pullbacks of the natural holomorphic symplectic forms on the total spaces of the canonical line bundles of complex curves. More generally, we show that any normal surface singularity is homeomorphic to a $\mathbb{Q}$-Gorenstein singularity whose index is equal to the smallest common denominator of the coefficients of the canonical cycle of the starting singularity.