The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Hölder-$\frac{2}{3}^-$ continuous paths. This work completes the construction of the central object in the Kardar–Parisi–Zhang universality class, the directed landscape.

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasi-projective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in $(C^\times)^n$. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory along the skeleton of the hypersurface. Using Mikhalkin–Viro patchworking, we identify the skeleton of the hypersurface with the boundary of the Fang–Liu–Treumann–Zaslow skeleton. By proving a new functoriality result for Bondal’s coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki’s recent theorem on mirror symmetry for toric varieties.

Sendov’s conjecture asserts that if a complex polynomial $f$ of degree $n \geqslant 2$ has all of its zeros in closed unit disk $\lbrace z: \vert z \vert \leqslant 1 \rbrace$, then for each such zero $\lambda_0$ there is a zero of the derivative $f^\prime$ in the closed unit disk $\lbrace z: \vert z-\lambda_0 \vert \leqslant 1 \rbrace$. This conjecture is known for $n \lt 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov’s conjecture holds for $n \geqslant n_0$. For $\lambda_0$ away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for $\lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $\lambda_0$ is extremely close to the unit circle); and for $\lambda_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.