arXiv research

Study TMF-valued TQFT for closed 3-manifolds using torsion linking pairings.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…

We propose a new method for studying $n$- and $Γ$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $Γ$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $Γ$-cohomology of…

Study deformations of compact Calabi-Yau conifolds with singularities.

Extended Rank-One Theorem to special metric spaces.

The paper reveals low-rank structure in neural network gradients, influenced by data and model parameters.

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

Constructs explicit p-harmonic functions on specific Lie groups.

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for Hilbert geometries and characterize them as being equivalent to contracting elem…

The study establishes uncertainty principles on harmonic manifolds of rank one.

Classifies foliations on specific symmetric spaces.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

Improved stability for matrix recovery from rank-one measurements.

Classifies rank-one submanifolds in Euclidean space.

No Einstein hypersurfaces found in Damek-Ricci spaces.

New insights into compact rank-one ECS manifolds, proving they are bundles over circles.

Compact rank one symmetric spaces are rigid under certain curvature conditions.

Study closed manifolds with rank one ray structures, proving completeness or covering properties.

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

Researchers describe the metric structure of compact ECS manifolds.

Stochastic Rank-One Bandits (Katarya et al, (2017a,b)) are a simple framework for regret minimization problems over rank-one matrices of arms. The initially proposed algorithms are proved to have logarithmic regret, but do not match the existing lower bound for this problem. We close this gap by first proving that rank…

Novel algorithm for Markov decision processes using rank-one approximation.

Rank-one measurements limit feasible sets for low-rank PSD matrices.

In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…

New random walk results on rank one symmetric spaces.

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.

The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.

Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.

Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.

We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.

We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.

New rigidity theorem for product of lattices.

New method unifies and formalizes data partitioning using a single vector.

Completes the space of vector-valued one-forms on manifolds.

We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…

Study shows how fast a specific matrix completion method works.

Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called…

This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.

There has been growing interest in extending traditional vector-based machine learning techniques to their tensor forms. An example is the support tensor machine (STM) that utilizes a rank-one tensor to capture the data structure, thereby alleviating the overfitting and curse of dimensionality problems in the conventio…

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetr…

We give an explicit description of all complete $G$-invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.