We investigate periodic diffeomorphisms of noncompact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with nonzero Euler characteristic. In the irreducible locally symmetric case, we show that no complete metric has more symmetry than the locally symmetric metric. In the moduli space case, we build on work of Farb and Weinberger and we prove an analogue of Royden’s theorem for complete finite volume metrics.

Matrices of rank at most $k$ are defined by the vanishing of polynomials of degree $k+1$ in their entries (namely, their $\left(\right(k+1)\times (k+1\left)\right)$-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each $k$ there exists an upper bound $d=d\left(k\right)$ such that tensors of border rank at most $k$ are defined by the vanishing of polynomials of degree at most $d$, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.

We give a new proof and an extension of the celebrated theorem of Hirzebruch and Zagier that the generating function for the intersection numbers of the Hirzebruch–Zagier cycles in (certain) Hilbert modular surfaces is a classical modular form of weight $2$. In our approach, we replace Hirzebruch’s smooth complex analytic compactification of the Hilbert modular surface with the (real) Borel–Serre compactification. The various algebro-geometric quantities that occur in the theorem are replaced by topological quantities associated to $4$-manifolds with boundary. In particular, the “boundary contribution” in the theorem is replaced by sums of linking numbers of circles (the boundaries of the cycles) in $3$-manifolds of type Sol (torus bundle over a circle) which comprise the Borel–Serre boundary.

In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider $N\times N$ symmetric Wigner matrices $H$ with $H_{ij}=N^{-1/2}{x}_{ij}$ whose upper-right entries $x_{ij}$ $(1\le i<j\le N)$ are independent and identically distributed (i.i.d.) random variables with distribution $\nu$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are i.i.d. random variables with distribution $\mathrm{\nu ̃}$. The means of $\nu$ and $\mathrm{\nu ̃}$ are zero, the variance of $\nu$ is 1, and the variance of $\mathrm{\nu ̃}$ is finite. We prove that the Tracy–Widom law holds if and only if ${lim\hspace{0.17em}}_{s\to \infty }{s}^4\mathbb{P}\left(\right|x_{12}|\ge s)=0$. The same criterion holds for Hermitian Wigner matrices.

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,\pi ,{Sym}^2\otimes \chi )$ of an irreducible cuspidal automorphic representation $\pi$ of ${GL}_r\left(\mathbb{A}\right)$ twisted by a Hecke character $\chi$. In particular, we will show that the $L$-function $L^S(s,\pi ,{Sym}^2\otimes \chi )$ is holomorphic for the region $Re\left(s\right)>1-1/r$ with the exception that, if ${\chi }^r{\omega }^2=1$, a pole might occur at $s=1$, where $\omega$ is the central character of $\pi$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $\chi =1$.