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We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set-theoretic point of view. This work is largely foundational in conception and character, recording both a framework for general…

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. The Milnor-Schwarz lemma implies that groups with a common model geometry are quasi-isometric; however, the…

This paper concerns the characterisation of second order marginals for random sets in a discrete setting. Under the instance of unit covariances, this problem possesses a combinatorial symmetry, exploited jointly in the companion paper to…

A real Bott manifold is the total space of an iterated $\RP ^1$-bundles over a point, where each $\RP^1$-bundle is the projectivization of a Whitney sum of two real line bundles. In this paper, we characterize real Bott manifolds which…

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex…

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an…

Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $\lambda(\mathcal{H}) > \binom{n-2}{r-1}$ must contain a Hamiltonian Berge cycle unless…

We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially.

We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2…

Consider a compact K\"{a}hler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $\lambda_1(\widetilde{M}%) =m^2.$ Then we prove that…

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random…

Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of H k (M ; Z). We prove that the image of this…

This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the…

By gluing some copies of a polytope of Kerckhoff and Storm's, we build the smallest known orientable hyperbolic 4-manifold that is not commensurable with the ideal 24-cell or the ideal rectified simplex. It is cusped and arithemtic, and has…

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed,…

One proves that there exists an obstruction to an open simply connected $n$-manifold of dimension $n\geq 5$ being geometrically simply connected. In particular there exist uncountably many simply connected $n$-manifolds which are not…

We study isometric immersions $f: M^n \rightarrow \mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal…

The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete K\"{a}hler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of…

The Steenrod problem for closed orientable manifolds was solved completely by Thom. Following this approach, we solve the Steenrod problem for closed orientable orbifolds, proving that the rational homology groups of a closed orientable…

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In this note, we first construct an $8$-dimensional compact smooth manifold whose unique…