We prove that the higher direct images $R^q{f}_{\ast }{\mathrm{\Omega }}_{\mathcal{Y}∕S}^p$ of the sheaves of relative Kähler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have k-Du Bois local complete intersection singularities, for $p\le k$ and all $q\ge 0$, generalizing a result of Du Bois (the case $k=0$). We then propose a definition of k-rational singularities extending the definition of rational singularities, and show that, if X is a k-rational variety with either isolated or local complete intersection singularities, then X is k-Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi–Yau varieties.

In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the k-rationality definition proposed here is equivalent to a previously given numerical definition for k-rational singularities. As an immediate consequence, it follows that for hypersurface singularities, k-Du Bois singularities are $(k-1)$-rational. This statement has recently been proved for all local complete intersection singularities by Chen, Dirks, and Mustaţă.

We prove that an infinite Riemann surface X is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are crosscuts is of zero measure. Then we establish the density of the Jenkins–Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff’s formula for the Teichmüller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard–Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.

We construct solutions of the 2D incompressible Euler equations in ${\mathbb{R}}^2\times [0,\mathrm{\infty })$ such that initially the velocity is in the super-critical Sobolev space $H^{\mathit{\beta }}$ for $1<\mathit{\beta }<2$, but is not in $H^{{\mathit{\beta }}^{\prime }}$ for ${\mathit{\beta }}^{\prime }>1+\frac{(3-\mathit{\beta })(\mathit{\beta }-1)}{2-{(\mathit{\beta }-1)}^2}$ for any $t\in (0,\mathrm{\infty })$. These solutions are not in the Yudovich class, but they exist globally in time and they are unique in a determined family of classical solutions.

We begin the study of categorifications of Donaldson–Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space, which we call DT categories. The DT category is defined to be the category of matrix factorizations on the noncommutative Hilbert scheme with a super-potential whose critical locus is the Hilbert scheme of points.

The first main result in this paper is the construction of semiorthogonal decompositions of DT categories, which can be regarded as categorical wall-crossing formulas of the framed triple loop quiver. Each summand is given by the categorical Hall product of some subcategories of matrix factorizations, called quasi-BPS categories. They are categories of matrix factorizations on twisted versions of noncommutative resolutions of singularities considered by Špenko and Van den Bergh, and were used by the first author to prove a PBW theorem for K-theoretic Hall algebras.

We next construct explicit objects of quasi-BPS categories via Koszul duality equivalences, and show that they form a basis in the torus localized K-theory. These computations may be regarded as a numerical K-theoretic analogue in dimension three of the McKay correspondence for Hilbert schemes of points. In particular, the torus localized K-theory of DT categories has a basis whose cardinality is the number of plane partitions, giving a K-theoretic analogue of MacMahon’s formula.