相关论文: Optimal Horoball Packing Densities for Koszul-type…
In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For…
Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a K\"{a}hler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete…
We study the global strong solutions to a 3-dimensional parabolic-hyperbolic Keller-Segel model with initial data close to a stable equilibrium with perturbations belonging to $L^2(\mathbb R^3)\times H^1(\mathbb{R}^3)$. We obtain global…
We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic…
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile…
Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.
We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since…
This paper provides an iterative procedure for constructing hyperbolic Coxeter groups that virtually fiber over $\mathbb{Z}$ that is flexible enough to yield infinitely many isomorphism classes in each virtual cohomological dimension (vcd)…
This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $\mu$ on $\mathbb{R}^d$), how many homothets of $C$…
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider…
It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher…
Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are…
In this paper, we investigate the relation between Sobolev-type embeddings of Haj{\l}asz-Besov spaces (and also Haj{\l}asz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,\mu)$ and lower bound for the measure $\mu.$ We…
This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each…
We compute Coxeter diagrams of several ``large'' reflective even 2-elementary hyperbolic lattices and their maximal parabolic subdiagrams, and give some applications of these results to the theory of K3 surfaces and hyperkahler varieties.
In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.
We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group…
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of…
Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint…
Let $T$ be an infinite volume Coxeter tetrahedron in three dimensional real hyperbolic space ${\bf H}^{3}_{\mathbb R}$ with two opposite right-angles and the other angles are all zeros. Let $G$ be the Coxeter group of $T$, so…