Abstract:
It is generally accepted that zero-growth population would be the long-term destiny of any population. China’s population is expected to reach 1.4 billion with zero-growth around 2030, if the low fertility policy continues up to then. Demographic dynamics indicate that the age composition of a steady zero-growth society would asymptotically approach the population mix of today’s many developed countries. Here we present a brief analysis and some insights into the age composition of a zero-growth society and the connectedness between total fertility rate, net reproduction rate and replacement level of fertility. Other formulas useful for demographic studies are also provided to further the analysis. Our results reveal that the age composition of China’s population in 2050 would be similar to those of some developed countries today. We argue that the misgivings about “population aging” or the fear of a “winter of humanity” in China stem from rather oversimplified estimations.

Abstract:
This is a passage excerpted from its author’s memoir written recently. The author recounts the role he played in the establishment of official diplomatic relation between the People’s Republic of China and Israel 20 years ago. The memoir cites the historical records that show a considerable Jewish community who migrated from Mumbai, India 1000 years ago and settled down in Kaifeng, the Capital of Song Dynasty (960-1127). The ensuing emperors, personally concerned about these Diasporas, allowed them to stay in the Capital, conferred Chinese surnames upon them, and made official positions in government obtainable by them. Synagogues had stood in Kaifeng for as long as 700 years until they were finally destroyed by a big flood in 1854. Since then they had never been restored. The Jewish community gradually dispersed nationwide and fully intermixed with other ethnicities via intermarriage. Now millions of Chinese people may have distant lineage from Kaifeng Jews. China is one of a few countries that have always been treating Diasporas graciously as compatriots for a millennium.

Abstract:
It is intended to find the best representation of high-dimensional functions or multivariate data inL 2(Ω) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear integral equations has been derived as an eigenvalue problem of gradient operator in the said space. It proved that the complete set of eigenfunctions generated by the gradient operator constitutes an orthonormal system, and any function ofL 2(Ω) can be expanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, the greatest eigenvalue of the integral operators has multiplicity 1 if the dimension of the underlying space n,n = 2, 4 and 6.

Abstract:
We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_{\mathbb{P}^m}(-a_j)$ for $a_j\in\mathbb{Z}^+$ and $\sum_{j=1}^{n-m} a_j \leq m$ in Gromov-Hausdorff topology with suitable initial Kahler class. We also show that the Kahler-Ricci flow resolves a family of isolated singularities uniquely in Gromov-Hausdorff topology. In particular, we construct global and local examples of metric flips by the Kahler-Ricci flow as a continuous path in Gromov-Hausdorff topology.

Abstract:
We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original projective variety, with well-defined tangent cones. We also prove a special degeneration for Kahler-Einstein manifolds of general type as an approach to establish the compactification of the moduli space of Kahler-Einstein manifolds of general type. A number of applications are given for degeneration of Calabi-Yau manifolds and the Kahler-Ricci flow on smooth minimal models of general type.

Abstract:
The global holomorphic \alpha-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the \alpha-invariant of toric Fano manifolds.

Abstract:
We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on a Kahler manifold with a polarized initial Kahler metric. We prove that the Kahler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kahler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth solutions of the Kahler-Ricci flow.

Abstract:
It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and $L^2$-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Riemannian structure on canonical models. Our approach can be viewed as the Kodaira embedding theorem on singular metric spaces with canonical Kahler metrics.