Related papers: Nearly Optimal Embeddings of Flat Tori

Using free random varaibles we find an embedding of the operator space $OH$ in the predual of a von Neumann algebra. The properties of this embedding allow us to determined the projection constant of $OH_n$, i.e. there exists a projection…

We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$,…

We prove that the distortion of any embedding into $L_1$ of the transportation cost space or earth mover distance over a $d$-dimensional grid $\{1,\dots m\}^d$ is $\Omega(\log N)$, where $N$ is the number of vertices and the implicit…

We give explicit origami embeddings of a 2-dimensional flat torus of any modulus in the 3-dimensional Euclidean space.

It has been shown by Indyk and Sidiropoulos [IS07] that any graph of genus g>0 can be stochastically embedded into a distribution over planar graphs with distortion 2^O(g). This bound was later improved to O(g^2) by Borradaile, Lee and…

We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a…

We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $\Omega(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil…

Dang et al. have given an algorithm that can find a Tarski fixed point in a $k$-dimensional lattice of width $n$ using $O(\log^{k} n)$ queries. Multiple authors have conjectured that this algorithm is optimal [Dang et al., Etessami et al.],…

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature…

We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that the local dimension of the first and middle layers of the $n$-dimensional Boolean lattice is asymptotically $\frac{n}{\log_2…

We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to…

A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A…

We study the algebra ${\cal A}_n$ and the basis of the Hilbert space ${\cal H}_n$ in terms of the $\theta$ functions of the positions of $n$ solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation…

We give an embedding of the Poincar\'e halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$…

The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this…

A general result by Jackson (Flat algebras and the translation of universal Horn logic to equational logic, J. Symb. Log. 73(1) (2008) 90--128) implies that the lattice of all quasivarieties of groups of exponent dividing $n$ embeds into…

We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong integrability condition on $V$, using the support of its Fourrier transform. We then use…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular…

Brendle proved Lawson conjecture about minimal embedded torus in the round three-dimensional sphere. Carlotto and Schulz constructed a minimal embedded three-dimensional hypertorus in the round four-dimensional sphere and conjectured that…