arXiv research

We discuss in this article a property of action of groups by isometries called "well displacing". An action is said to be well displacing, if the displacement function is equivalent to the the displacement function for the action on the Cayley graph. We relate this property with the fact that orbit maps are quasi-isome…

Study automorphism groups of quandles up to order 10.

New displacement technique vanishes bounded cohomology in all degrees.

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the $\log (2k-1)$ Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as …

Study metrics on quandles, a knot theory algebraic system.

Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded…

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $Γ$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds fo…

Study of conjugacy classes in infinite-type surfaces' mapping class groups.

In this paper we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic manifold. Essential to the proofs is a displacement principle for such submanifolds. Namely, we show that a topologicall…

For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $V(γ)=e^{λA(γ) i}$ where $A(γ)$ is …

PIE-PINN estimates elastic properties from noisy, low-res displacement data.

Nonnegative sectional curvature linked to matrix displacement convexity.

We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hofer-like norm [5] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positiv…

The Homogeneity Conjecture explores if constant displacement isometries imply homogeneous spaces.

Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.

This paper optimizes functions of probability measures using particle gradient descent for displacement convex functions.

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on…

Study group actions on hyperbolic spaces to find algebraic and geometric properties.

Random walks on hyperbolic spaces follow predictable large deviation principles.

This paper explores the possibility that asset prices, especially those traded in large volume on public exchanges, might comply with specific physical laws of motion and probability. The paper first examines the basic dynamics of asset price displacement and finds one can model this dynamic as a harmonic oscillator at…

The low displacement rank (LDR) framework for structured matrices represents a matrix through two displacement operators and a low-rank residual. Existing use of LDR matrices in deep learning has applied fixed displacement operators encoding forms of shift invariance akin to convolutions. We introduce a class of LDR ma…

Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

New framework uses symmetry-based matrices for efficient, flexible NNs.

New curvature measure for optimal transport with specific cost function.

Training models to prefer certain responses can unintentionally shift probability to harmful ones.

We prove that the displacement energy of a stable coisotropic submanifold is bounded away from zero if the ambient symplectic manifold is closed, rational and satisfies a mild topological condition.

The paper explores metrics on Lie groups and their connections to dual quaternions.

The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.

This paper proposes a representational model for image pairs such as consecutive video frames that are related by local pixel displacements, in the hope that the model may shed light on motion perception in primary visual cortex (V1). The model couples the following two components: (1) the vector representations of loc…

In this paper, it is shown that the reduced $\ell^p$-cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exist a sequence of measures $ξ_n$ converging to a left-invariant mean and such that the transport cost between $ξ_n$ displaced b…

We study Hitchin representations and maximal symplectic representations of surface groups, which can be both thought of as generalisations of Fuchsian representations. We show that the corresponding energy functionals are proper on Teichmuller space. We also prove that the mapping class group acts properly on the corre…

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space $P(M)$ of probability measures over $M$. We show that $M$ has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if and only…

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…

In this paper we make the first steps towards developing a theory of intersections of coisotropic submanifolds, similar to that for Lagrangian submanifolds. For coisotropic submanifolds satisfying a certain stability requirement we establish persistence of coisotropic intersections under Hamiltonian diffeomorphisms, ak…

The paper studies the action of automorphisms on train tracks and finds that the set of minimally displaced points is co-compact.

Paper derives formulas for surface variations in shell theory.

Armed conflict has led to an unprecedented number of internally displaced persons (IDPs) - individuals who are forced out of their homes but remain within their country. IDPs often urgently require shelter, food, and healthcare, yet prediction of when large fluxes of IDPs will cross into an area remains a major challen…

Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.

We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: Any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points. Moreover, any Stein …

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $Ø$ and any point $x \in Ø$, any discrete group generated by a finite number of automorphisms of $Ø$, which displace $x$ at …

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …

We show that if the monodromy of an open book decomposition has sufficiently high displacement distance, acting on the loop and arc complex for a page, then it is the unique minimal Euler characteristic open book for the manifold. In particular, we show that such an open book induces the unique (up to isotopy) minimal …

Given an affine isometry of $\R^3$ with hyperbolic linear part, its Margulis invariant measures signed Lorentzian displacement along an invariant spacelike line. In order for a group generated by hyperbolic isometries to act properly on $\R^3$, the sign of the Margulis invariant must be constant over the group. We show…

Classification of groups as symmetries of infinite translation surfaces.

Recently low displacement rank (LDR) matrices, or so-called structured matrices, have been proposed to compress large-scale neural networks. Empirical results have shown that neural networks with weight matrices of LDR matrices, referred as LDR neural networks, can achieve significant reduction in space and computation…

In this paper, we propose a nonlinear distance metric learning scheme based on the fusion of component linear metrics. Instead of merging displacements at each data point, our model calculates the velocities induced by the component transformations, via a geodesic interpolation on a Lie transfor- mation group. Such vel…