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Consider two random vectors $\wt{\bx} \in \R^p$ and $\wt{\by} \in \mathbb R^q$ of the forms $\wt{\bx}=A \mathbf z + \bC_1^{1/2}\mathbf x $ and $\wt{\by}=B \mathbf z + \bC_2^{1/2}\mathbf y $, where $\mathbf x \in \R^p$, $\mathbf y \in \R^q$ and $\mathbf z\in \R^r$ are independent random vectors with i.i.d. entries of zero mean and unit variance, $\bC_1$ and $\bC_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices, and $A$ and $B$ are $p \times r$ and $q\times r$ deterministic factor loading matrices. With $n$ independent observations of $(\wt{\mathbf x},\wt{\mathbf y})$, we study the sample canonical correlations between $\wt{\bx} $ and $\wt{\by}$. We consider the high-dimensional setting with finite rank correlations, that is, ${p}/{n}\to c_1$ and ${q}/{n}\to c_2$ as $n\to \infty$ for some constants $c_1\in (0,1)$ and $c_2\in (0,1-c_1)$, and $r$ is a fixed integer. Let $t_1\ge t_2 \ge \cdots\ge t_r\ge 0$ be the squares of the nontrivial population canonical correlation coefficients between $\wt {\bx}$ and $\wt{\by}$, and let $\wt\lambda_1 \ge \wt\lambda_2\ge \cdots \ge \wt\lambda_{p\wedge q}\ge 0$ be the squares of the sample canonical correlation coefficients. If the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$ are i.i.d. Gaussian, then the following dichotomy has been shown in \cite{CCA} for a fixed threshold $t_c \in(0, 1)$: for any $1\le i \le r$, if $t_i < t_c$, then $\wt\lambda_i$ converges to the right-edge $\lambda_+$ of the limiting eigenvalue spectrum of the sample canonical correlation matrix, and moreover, $n^{2/3}(\wt\lambda_i-\lambda_+)$ converges weakly to the Tracy-Widom law; if $t_i>t_c$, then $\wt\lambda_i$ converges to a deterministic limit $\theta_i \in (\lambda_+, 1)$ that is determined by $c_1$, $c_2$ and $t_i$. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate $t_i$'s and for $t_i$'s that are close to the threshold $t_c$. Finally, we also provide almost sharp convergence rates for the sample canonical correlation coefficients under a general $a$-th moment assumption.

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We propose some conjectures on the generating series of (equivariant) Euler characteristics of some vector bundles constructed from the tautological bundles on Hilbert schemes of points on affine $k$-spaces. We establish the surface case of these conjectures and present some verifications of the higher dimensional cases.

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two-and three-dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge’s formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.