In this lecture, we will introduce the basic definitions and summarize some main results on the value distribution theory of algebroid functions. The results focus on the singular directions, filling discs, uniqueness theorems. These results are compared with the case of meromorphic functions. Some open problems are also proposed.

Random genetic drift occurs at a single unlinked locus with two or more alleles. The probability density of alleles is governed by a degenerated Fokker-Planck equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary as time evolves, which is just the so-called fixation phenomenon. In order to find a complete solution which should keep the conservation of positivity, total probability and expectation, different schemes of FDM, FVM and FEM are tested to solve the equation numerically. We observed that the methods have totally different long-time behaviors. Some of them are stable and keep the conservation of positivity and probability, but fail to keep the expectation. Some of them fails to keep the positivity.

In this talk, we will revisit the Heterogeneous Multiscale Methods(HMM) for homogenization of elliptic problems. Firstly we talk about the relationship between HMM and other multiscale methods such as Variational Multiscale Methods, Multiscale Finite Element Methods and Free Bubble Methods. Then we discuss the error analysis for problems without scale separation.

The efficiency of estimation for the parameters in semiparametric models has been widely studied in the literature. In this paper, we study efficient estimators for both parameters and nonparametric functions in a class of generalized semi/non-parametric regression models, which cover commonly used semiparametric models such as partially linear models, partially linear single index models, and two-sample semiparametric models. We propose a maximum likelihood principle combined with the local linear technique for estimating the parameters and nonparametric functions. The proposed estimators of the parameters and a linear functional of the nonparametric functions are consistent and asymptotically normal and are further shown to be semiparametrically efficient.

In this paper, we consider prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in hyperbolic space. Under some sufficient conditions, we obtain an existence result by the standard degree theory based on a priori estimates for solutions to the equations. Different from the prescribed Weingarten curvature problem in space forms, we do not impose a condition for radial derivative of the functions in the right hand side of the equations to prove the existence due to the horo-convexity of hypersurfaces in hyperbolic space.

Based on reverse isoperimetric inequalities on Grassmann manifolds, a Grassmannian Loomis-Whitney inequality and its dual are established, which provides a lower bound for the volume of an origin-symmetric convex body in terms of its lower dimensional sections.

The celebrated Legendre ellipsoid and the LYZ ellipsoid introduced by Lutwak, Yang, and Zhang in 2000 are important concepts in convex geometric analysis. These ellipsoids are generated by the cosine transform (i.e., this transform origins from the inner product of two vectors). In this talk, we will discuss two types of ellipsoids by using the sine transform (i.e., this transform is related to the cross product of two vectors), which can be considered as the sine counterparts of the Legendre ellipsoid and the LYZ ellipsoid.

In this talk, we investigate the evolution of a curvature flow in the plane, which can be regarded as the gradient flow of isoperimetric ratio, for immersed locally convex closed curves. In particular, it is shown that the flow evolves two classes of rotationally symmetric curves, i.e., highly symmetric curves and Abresch-Langer type curves, into $m$fold circles as time goes to infinity.