We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal{A}$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval $(0; 1/4]$, then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits a $C^1$-smooth first integral with non-vanishing gradient on $\mathcal{A}$, then the curve $\gamma$ is an ellipse.

The main ingredients of the proof are (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a ``background" solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik (2018). The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák (2015). Our solutions live precisely on the borderline of the known well-posedness theory.