We give a new proof for the existence of mean curvature flow with surgery of $2$-convex hypersurfaces in ${\mathbb{R}}^N$. Our proof works for all $N\ge 3$, including mean convex surfaces in ${\mathbb{R}}^3$. We also derive a priori estimates for a more general class of flows in a local and flexible setting.

We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application, we consider the critical wave equation and prove the asymptotic stability of the ordinary differential equations blowup profile in the energy space.

It is shown that the maximum of $\left|\zeta \right(1/2+it\left)\right|$ on the interval $T^{1/2}\le t\le T$ is at least $exp\left(\right(1/\sqrt{2}+o\left(1\right)\left)\sqrt{logTlogloglogT/loglogT}\right)$. Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant $A$ in the inequality

$${sup\hspace{0.17em}}_{1\le n_1<\cdots <n_N}{\sum }_{k,\ell =1}^N\frac{gcd(n_k,n_{\ell })}{\sqrt{n_k{n}_{\ell }}}\ll Nexp\left(A\sqrt{\frac{logNlogloglogN}{loglogN}}\right),$$ established in a recent paper of ours, cannot be taken smaller than $1$.

In this article we obtain sharp obstructions to the symplectic embedding of the Lagrangian bidisk into four-dimensional balls, ellipsoids, and symplectic polydisks. We prove, in fact, that the interior of the Lagrangian bidisk is symplectomorphic to a concave toric domain by using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the Lagrangian bidisk.

We prove that the eigencurve associated to a definite quaternion algebra over $\mathbb{Q}$ satisfies the following properties, as conjectured by Coleman and Mazur as well as Buzzard and Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components, each finite and flat over the weight annuli; (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space; and (c) the sequence of the slope ratios forms a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves toward the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.