It is shown that there exist a sequence of $3$-regular graphs $\{G_n{\}}_{n=1}^{\infty }$ and a Hadamard space $X$ such that $\{G_n{\}}_{n=1}^{\infty }$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of the second author and Silberman. The graphs $\{G_n{\}}_{n=1}^{\infty }$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody algebra of type $A_n^{\left(1\right)}$ ( $n\ge 1$) or $D_n^{\left(1\right)}$ ( $n\ge 4$), and let ${\mathfrak{g}}_0$ be the underlying finite-dimensional simple Lie subalgebra of $\mathfrak{g}$. For each Dynkin quiver $Q$ of type ${\mathfrak{g}}_0$, Hernandez and Leclerc introduced a tensor subcategory ${\mathcal{C}}_Q$ of the category of finite-dimensional integrable $U{\text{'}}_q\left(\mathfrak{g}\right)$-modules and proved that the Grothendieck ring of ${\mathcal{C}}_Q$ is isomorphic to $\mathbb{C}\left[N\right]$, the coordinate ring of the unipotent group $N$ associated with ${\mathfrak{g}}_0$. We apply the generalized quantum affine Schur–Weyl duality to construct an exact functor $\mathcal{F}$ from the category of finite-dimensional graded $R$-modules to the category ${\mathcal{C}}_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to ${\mathfrak{g}}_0$. We prove that the homomorphism induced by the functor $\mathcal{F}$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\mathcal{F}$ sends the simple modules to the simple modules.

We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.

Building on the work of Goto, Nishida, and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with $n$ punctures is not a Mori dream space for $n>133$. This answers a question posed by Hu and Keel.