arXiv research

Paper proves a noncompact version of Gromov's band-width estimate.

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

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

Derives generalizations of the long neck principle and spectral width inequality.

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds $(M, ω)$ with $b_2(M)=1$. As an application we obtain an upper bound on the Seshadri constant $ε(L)$ where $L$ is the ample line bundle on $M$ such that $c_1(L)=[\fracωπ]$.

Estimates bandwidth for CMC initial data sets.

Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also c…

Proves a quantitative index theorem for positive scalar curvature metrics.

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

Study on Klein bottle's cotangent bundle using contact homology.

Extends width estimates to family case using index theory.

The paper proves rigidity theorems for area widths of Riemannian manifolds.

Study on Gehring link problem and width of bands in curved manifolds.

In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to $6$ and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar curvature metrics due to Gromov and Lawson. We develop analogies between the area of…

The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.

We study the number of Darboux charts needed to cover a closed connected symplectic manifold $(M,ω)$, and effectively estimate this number from below and from above in terms of the Lusternik--Schnirelmann category of $M$ and the Gromov width of $(M,ω)$.

A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters bounded by c. We prove that an n-dimensional Riemannian manifold M with at least n-k…

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width inequalities for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If $M$ is filling enlargeable…

The paper explores index theory for Dirac operators to understand scalar curvature properties.

The study refines known counterexamples in 4D to satisfy certain inequalities.

Random groups prove length constraints on product of conjugates.

Develops connections between operator K-theory and positive scalar curvature.

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with va…

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \…

The paper proves rigidity for certain product spaces and bounds for band widths.

An action selector associates, in a suitable way, to each compactly supported Hamiltonian on a symplectic manifold an action value of the Hamiltonian. Action selectors are known to exist for a broad class of symplectic manifolds. We show how the existence of an action selector leads to sharp energy capacity inequalitie…

The paper establishes distance estimates for manifolds with lower scalar curvature bounds.

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…

Sharp inequalities for Kähler manifolds' systolic invariants are established.

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in w…

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly sque…

The paper confirms a conjecture about manifolds with positive curvature.

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.

The isospectral problem for p-widths is solved using Zoll metrics on S^2.

Width trees link link invariants and bridge number.

Lectures on deep learning properties in infinite and large-width networks.

Computed p-widths for hemisphere, first for manifolds with boundary.

Polygon $p$-widths are found via billiard trajectories.

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…

Computed p-widths for real projective plane.

Study bounds Urysohn width of manifolds under surgeries.

Residual networks with block width max(d_x, d_y) approximate all functions.

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…

Proves conjecture about sphere widths under rotational symmetry.

New link invariants from diagram colorings match link widths.

Study infinite-depth limits of neural networks with fixed width.

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we …