探索性数据分析对于建立合理的统计模型、使用恰当的统计推断方法以及验证统计模型和统计推断所需要的假设条件至关重要。对于低维数据，探索性数据分析的方法和理论发展较为成熟。但是，对于高维数据而言，发展探索性数据分析方法和理论非常困难。对高维数据进行降维，是发展探索性高维数据分析方法和理论的合理选择。高维数据降维需要解决两个非常核心的科学问题：如何刻画数据的非线性相依关系以及如何在不假设任何模型的前提下对高维数据进行降维。不管使用哪种降维方法，一个合理的要求是希望降维以后的数据还能完整包含原始高维数据所包含的信息。我们将系统讨论这两个科学问题的可能解决办法。

This paper studies two-sample functional linear regressions with functional responses, where the regression functions are assumed to have a scaling transformation. We estimate the intercept function, slope function, and parameter components based on the least squares method and functional principal component analysis. The proposed estimator of the parameter components are shown to be root-n consistent and asymptotically normal.

这是一个综述报告。我们将介绍最近几年来离散几何分析，主要是图上的几何分析的研究进展。

In this talk, we present multi-continua upscaling methods for multiscale problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are intended for problems without scale separation and high contrast, which often occur in applications. In our method, the local solutions are used as a forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem.

The asymptotic distribution theory for coefficients of a polynomial is an active topic in asymptotic analysis. In 1967, Harper proposed a criterion to measure the asymptotic normality of a series of numbers, when he researched the asymptotic behavior of Stirling numbers of the second kind. In this talk, we will discuss some further asymptotic normality criteria of coefficients of a polynomial with all real roots or purely imaginary roots (including 0). These new asymptotic normality criteria turn out to be very efficient and have abundant applications in combinatorics, mainly including the coefficients of a series of characteristic polynomials of adjacency matrix, Laplacian matrix, signless Laplacian matrix, skew-adjacency matrix, chromatic polynomial

New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal- continuous Lagrange elements of any order on simplexes in two- and three- dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, by establishing the key property of discrete compactness, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations.

In this talk, I will introduce the Finsler geometry and the background of the isoperimetric problem. By using the variational theory, we give the local solution of the isoperimetric problem in 2-dimensional Finsler space forms. This is the joint work with Mengqing Zhan.

In this talk, I first recalled some knowledge of the Kahler-Ricci flow. Then I will talk about the behavior of the Kahler-Ricci flow on some Fano bundles which is a trivial bundle on one Zariski open set. We show that if the fiber is $/mathbb{P}^{m}$ blown up at one point and the initial metric is in a suitable kahler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base. In the talk, I will aim to talk about in the case of product manifolds. This is a joint work with Xin Fu.