We construct examples of 4-dimensional manifolds $M$ supporting a locally CAT(0)-metric, whose universal covers $\mathrm{M̃}$ satisfy Hruska’s isolated flats condition, and contain $2$-dimensional flats $F$ with the property that ${\partial }^{\infty }F\cong S^1\hookrightarrow S^3\cong {\partial }^{\infty }\mathrm{M̃}$ are nontrivial knots. As a consequence, we obtain that the group ${\pi }_1\left(M\right)$ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonpositive sectional curvature. In particular, if $K$ is any compact locally CAT(0)-manifold, then $M\times K$ is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

In this article we prove a differentiable rigidity result. Let $(Y,g)$ and $(X,g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n⩾3$), and let $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\epsilon :=\epsilon (n,d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-(n-1)g$ and its volume satisfies ${vol}_g\left(Y\right)⩽(1+\epsilon ){vol}_{g_0}\left(X\right)$, then the manifolds are diffeomorphic. The proof relies on Cheeger–Colding’s theory of limits of Riemannian manifolds under lower Ricci curvature bound.

We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove sharp results for this class of discrete operators in all dimensions, providing necessary and sufficient conditions for them to extend to bounded operators from ${\ell }^p$ to ${\ell }^q$. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.

We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the number of representations of dimension at most $n$ is bounded by a low-degree polynomial in $n$. As a consequence, we show that the number of conjugacy classes of maximal subgroups of a finite almost simple group $G$ is at most $O\left(\right(log\left|G\right|{)}^3)$.

In this paper we study the Weil–Petersson geometry of $\overline{{ℳ}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{{ℳ}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C\sqrt{g}logg$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{{ℳ}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates.