The well-posedness of the dynamic framework in earth-system model (ESM for short) is a common issue in earth sciences and mathematics. In this paper, the authors first introduce the research history and fundamental roles of the well-posedness of the dynamic framework in the ESM, emphasizing the three core components of ESM, i.e., the atmospheric general circulation model (AGCM for short), land-surface model (LSM for short) and oceanic general circulation model (OGCM for short) and their couplings. Then, some research advances made by their own research group are outlined. Finally, future research prospects are discussed.

Let E be a holomophic vector bundle over a compact Astheno-Kähler manifold (M, ω). The authors would prove that E is a numerically flat vector bundle if E is pseudo-effective and the first Chern class $c_1^{BC}$ (E) is zero.

This paper is a survey of some recent developments concerning extremal Kähler metrics on Toric Manifolds.

In this paper, the authors provide a brief introduction of the path-dependent partial di.erential equations (PDEs for short) in the space of continuous paths, where the path derivatives are in the Dupire (rather than Fréchet) sense. They present the connections between Wiener expectation, backward stochastic di.erential equations (BSDEs for short) and path-dependent PDEs. They also consider the well-posedness of path-dependent PDEs, including classical solutions, Sobolev solutions and viscosity solutions.

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of ℝℙ ⁿ, Comm. Anal. Geom., 25, 2017, 799–846] initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space ℝℙ ^m. The purpose of this paper is to develop the work due to Pipoli and Sinestrari, and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space. Namely, the authors prove that if the initial submanifold in ℝℙ ^m satisfies a suitable pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t → ∞. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

On metrics of Eguchi-Hanson type II with negative constant Ricci curvatures, the authors show that there is no nontrivial Killing spinor. On metrics of Eguchi-Hanson type II with negative constant scalar curvature, they show that there is no nontrivial L ^p eigenspinor for 0 < p < 2 if the eigenvalue has nontrivial real part, and no nontrivial L ² eigenspinor if either the eigenvalue has trivial real part or the eigenvalue is real, the eigenspinor is isotropic and the parameter η in radial and angular equations for eigenspinors is real. They also solve harmonic spinors and eigenspinors explicitly on metrics of Eguchi-Hanson type II with certain special potentials.

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard’s sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether they can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, the authors will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order.

The authors introduce the conception of stationary map-varifold pairs and prove a compactness result. As applications, they analyse the asymptotic structure of the pseudo tangent map of stationary harmonic maps. For stationary pair, they also get a strong convergence criterion about the map part and introduce the stratification of the singular set.

Most of current public key cryptosystems would be vulnerable to the attacks of the future quantum computers. Post-quantum cryptography offers mathematical methods to secure information and communications against such attacks, and therefore has been receiving a significant amount of attention in recent years. Lattice-based cryptography, built on the mathematical hard problems in (high-dimensional) lattice theory, is a promising post-quantum cryptography family due to its excellent efficiency, moderate size and strong security. This survey aims to give a general overview on lattice-based cryptography. To this end, the authors begin with the introduction of the underlying mathematical lattice problems. Then they introduce the fundamental cryptanalytic algorithms and the design theory of lattice-based cryptography.