arXiv research

Paper answers Gromov's compactness question on noncompact manifolds.

Study on simplicial volume and Euler characteristic of aspherical manifolds.

Extends Gromov non-squeezing to locally conformally symplectic structures.

Proves a quantitative index theorem for positive scalar curvature metrics.

A well known question of Gromov asks whether every one-ended hyperbolic group $Γ$ has a surface subgroup. We give a positive answer when $Γ$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when…

The moduli space of isometry classes of Riemannian structures on a smooth manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum gravity. A natural question concerning it is: What is a natural topology on such moduli space that reflects best quantum fluctuations of the geometries within the Planc…

Bounds and constructions for Gromov-Hausdorff distance between spheres.

Proves counterexamples to Donaldson's 4-6 question and related conjectures.

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…

New theorem on spheres with punctures using infinity metric.

Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.

Study shows local topologies of certain geometric spaces.

The article disproves a local systolic inequality and shows a lower bound on filling area.

A theorem on odd dimensional noncompact manifolds shows curvature bounds.

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Grom…

Study curvature and symplectic properties of symmetric products of surfaces.

Recover simple irreversible Finsler geometry from travel time data

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk i…

New method uses cohomology to quantify molecular similarity.

The paper studies 3D manifolds with positive scalar curvature and volume growth.

The paper examines properties of GW optimal transport plans, showing they can be sparse and permutation-supported.

Proves non-existence of certain metrics on specific manifolds.

Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We pr…

Study shows different fundamental groups for manifolds with same limit.

Study shows rigidity of polyhedrons in hyperbolic spaces.

We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M, we construct a secondary invariant tau_(2):…

We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

Given any finite subset X of the sphere S^n, n>1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R^{n+1} whose Gauss map misses X. In particular, this answers a question of M. Gromov.

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of the…

In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.

Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot $T_{p,q}$ satisfies $δ(T_{p,q})>\frac 1{160}\min(p,q)$. This answers a 1983 question of Gromov.

New method connects CAT(0) spaces to hyperbolic spaces.

We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse …

Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This genera…

This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate t…

Study of groups and their quasi-isometrically embedded subgroups.

We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.

Paper bounds the A-hat genus using curvature and isoperimetric constants.

The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.

We show that a map with Hölder exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov fo…

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer …

Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.

We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively cur…

We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r$ and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $\log |TV_r (M)|$ is bounded above by a function linear in $r$ and whose slope is a positiv…

In this note we determine all possible dominations between different products of manifolds, when none of the factors of the codomain is dominated by products. As a consequence, we determine the finiteness of every product-associated functorial semi-norm on the fundamental classes of the aforementioned products. These r…

We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…

Paper introduces a new sampler for simulation-based inference using Gromov-Monge distance.