Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in $L$ through degeneration. We obtain a conjectural relationship between the potential function of $L$ and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the Seidel elements with the potential function.

We shall give a necessary and sufficient condition for the existence of stable sheaves on Enriques surfaces based on results of Kim, Yoshioka, Hauzer and Nuer. For unnodal Enriques surfaces, we also study the relation of virtual Hodge “polynomial” of the moduli stacks.

We establish the atomic decompositions for the weighted Hardy spaces with variable exponents. These atomic decompositions also reveal some intrinsic structures of atomic decomposition for Hardy type spaces.

The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.

We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with random initial data below the energy space.