Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.

We consider compact submanifolds of dimension n  2 in Rn+k, with nonzero mean curvature vector everywhere, and such that the full norm of the second fundamental form is bounded by a fixed multiple (depending on n) of the length of the mean curvature vector at every point. We prove that the mean curvature flow deforms such a submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in some (n + 1)- dimensional affine subspace of Rn+k.

In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Amp`ere equation modeled on those of Ben Andrews and Guti´errez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.

Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .