The Turán problem asks for the largest number of edges in an n-vertex graph not containing a fixed forbidden subgraph F. We construct a new family of graphs not containing $K_{s,t}$, for $t=C^s$, with $\mathrm{\Omega }(n^{2-1∕s})$ edges matching the upper bound of Kövári, Sós, and Turán.

Fix a Calabi–Yau 3-fold X satisfying the Bogomolov–Gieseker conjecture of Bayer, Macrì, and Toda, such as the quintic 3-fold. We express Joyce’s generalized DT invariants counting Gieseker semistable sheaves of any rank $r\ge 1$ on X in terms of those counting sheaves of rank 0 and pure dimension 2.

The basic technique is to reduce the ranks of sheaves by replacing them by the cokernels of their Mochizuki/Joyce–Song pairs and then using wall-crossing to handle their stability.

We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over ${\mathbb{F}}_q$) to Khovanov–Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $\mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.

We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging, and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli’s equation, we uncover an equivalent formulation as “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly, downstream waves are investigated in more detail.