We consider a general class of symmetric or Hermitian random band matrices $H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket^d}$ in any dimension $d\ge 1$, where the entries are independent, centered random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. We assume that $s_{xy}$ vanishes if $|x-y|$ exceeds the band width $W$, and we are interested in the mesoscopic scale with $1\ll W\ll N$. Define the {\it{generalized resolvent}} of $H$ as $G(H,Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix with entries $Z_{xx}\in \mathbb C_+$ for all $x$. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $d\ge 2$. More precisely, for any fixed $d\ge 2$, we prove that the bulk eigenvectors of $H$ are delocalized in certain averaged sense if $N\le W^{1+\frac{d}{2}}$. This improves the corresponding results in \cite{HeMa2018} under the assumption $N\ll W^{1+\frac{d}{d+1}}$, and in \cite{ErdKno2013,ErdKno2011} under the assumption $N\ll W^{1+\frac{d}{6}}$. For 1D random band matrices, our fluctuation averaging result was used in \cite{PartII,PartI} to prove the delocalization conjecture and bulk universality for random band matrices with $N\ll W^{4/3}$.

Big Data bring new opportunities to modern society and challenges to data scientists. On the one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This paper gives overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasize on

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to S4 or RP4 or quotients of S3 ×R by a cocompact fixed point free subgroup of the isometry group of the standard metric of S3 × R, or a connected sum of them.

The free-living SAR11 clade is a globally abundant group of oceanic Alphaproteobacteria, with small genome sizes and rich genomic A+T content. However, the taxonomy of SAR11 has become controversial recently. Some researchers argue that the position of SAR11 is a sister group to Rickettsiales. Other researchers advocate that SAR11 is located within free-living lineages of Alphaproteobacteria. Here, we use the natural vector representation method to identify the evolutionary origin of the SAR11 clade. This alignment-free method does not depend on any model assumptions. With this approach, the correspondence between proteome sequences and their natural vectors is one-to-one. After fixing a set of proteins, each bacterium is represented by a set of vectors. The Hausdorff distance is then used to compute the dissimilarity distance between two bacteria. The phylogenetic tree can be reconstructed based on these distances. Using our method, we systematically analyze four data sets of alphaproteobacterial proteomes in order to reconstruct the phylogeny of Alphaproteobacteria. From this we can see that the phylogenetic position of the SAR11 group is within a group of other free-living lineages of Alphaproteobacteria.

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