I would like to congratulate Professor Antoniadis for successfully outlining the current state-of-art of wavelet applications in statistics. Since wavelet techniques were introduced to statistics in the early 90s, the applications of wavelet techniques have mushroomed. There is a vast forest of wavelet theory and techniques in statistics and one can find himself easily lost in the jungle. The review by Antoniadis, ranging from linear wavelets to nonlinear wavelets, addressing both theoretical issues and practical relevance, gives in-depth coverage of the wavelet applications in statistics and guides one entering easily into the realm of wavelets.

We suggest that exotic sphere fibrations can be mapped to band topologies in condensed matter systems. These fibrations can correspond to geometric phases of two double bands or state vector bases with second Chern numbers m+n and -n, respectively. They can be related to topological insulators, magnetoelectric effects, and photonic crystals with special edge states. We also consider time-reversal symmetry breaking perturbations of topological insulator, and heterostructures of topological insulators with normal insulators and with superconductors. We consider periodic TI/NI/TI/NI heterostructures, and periodic TI/SC/TI/SC heterostructures. They also give rise to models of Weyl semimetals which have thermal and electrical transports.

We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption $p N \gg N^{2/3}$, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least $4 + \epsilon$ moments.

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