On Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

Improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data. Both involve estimation of the covariance function. Yet challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. A class of semiparametric models for the covariance function by that imposes a parametric correlation structure while allowing a nonparametric variance function is proposed. A kernel estimator for estimating the nonparametric variance function is developed. Two methods for estimating parameters in the correlation structurea quasi-likelihood approach and a minimum generalized variance methodare proposed. A semiparametric varying coefficient partially linear model for longitudinal data is introduced, and an estimation procedure for model coefficients using a profile weighted least squares approach

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17,19,6].

Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type ADE singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.

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