arXiv research

Proves inequality for 1-dimensional cycles.

Study confirms a 2-sphere metric with three geodesics of minimal length.

We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich-Lishak-Nabutovsky-Rotman. We show also that for any $C>0$ there is a Riemannian metric $g$ on a 3-sphere such that $\text{vol}(S^3,g)=1$ and for an…

Alternative proof of simplicial volume bound using area-minimizing sets.

The study shows conditions for larger volumes in the universal cover of a manifold.

This is an appendix to arXiv:1610.04888, "Quantitative null-cobordism", which improves one of the main results of that paper to a near-sharp one. It is not a self-contained paper.

We suggest several mathematical counterparts to the idea of "effective degrees of freedom" and formulate specific questions, much of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in bal…

We show that closed arithmetic hyperbolic n-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find…

We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized G…

We introduce a $\mathbb{Z}$--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homolog…

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

Proves spectral sequence for real Heegaard Floer homology.

We extend a systolic inequality of Guth for Riemannian manifolds of maximal $\mathbb{Z}_2$ cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface grou…

Proves the Weyl law for 1-cycles in manifolds.

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8…

We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each…

In this paper we prove the following. Let $Σ$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $Σ\times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\widetilde{Σ\times \mathbb{S}^1},\widetilde{g})$, we get a lower bound on th…

Quantum codes with optimal distance and dimension for n-dimensional space.

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…

Real bordered Floer homology computes 3-manifolds with involution.

In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarn…

The paper proves conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.

This paper presents connections between Gromov's work on isoperimetry of waists and Milman's work on the $M$-ellipsoid of a convex body. It is proven that any convex body $K \subseteq \mathbb{R}^n$ has a linear image $\tilde{K} \subseteq \mathbb{R}^n$ of volume one satisfying the following waist inequality: Any continu…

The paper shows equidistribution of geodesics and nets on manifolds.

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inve…

The paper classifies certain 4D and 5D manifolds with positive scalar curvature.

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

New metric properties show volume constraints in collapsing spaces.

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a resul…

Proves lower bounds on Hausdorff dimension of projections of invariant sets.

If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are related by the following isosystolic inequality: Sys(M,g)^n \leq n! Vol(M,g). The …

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, polynomial scaling distance and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contr…

The paper proves the existence of $G$-invariant minimal hypersurfaces on certain Riemannian manifolds.

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$ where $HC_m$ denotes the $m$-dimensional Hausdorff content,…

Estimates for graph embeddings into symmetric spaces derived from coarse geometry.

Generative models converge to data distribution but not principal latent factors.

The study quantifies topological expansion properties of complexes and their embeddings.